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\title{The inertial regimes in Rayleigh--Bénard convection}
\alttitle{Les régimes inertiels de la convection de Rayleigh--Bénard}

\author{\firstname{Bernard} \lastname{Castaing}}
\address{Laboratoire des écoulements géophysiques et industriels, Domaine Universitaire, CS 40700, 38058 Grenoble Cedex 9, France}

\author{\firstname{Francesca} \lastname{Chillà}\IsCorresp}
\address{ENSL, CNRS, Laboratoire de physique, 69342 Lyon, France}
\email{francesca.chilla@ens-lyon.fr}

\author{\firstname{Julien} \lastname{Salort}\CDRorcid{0000-0003-3052-5898}}
\address[2]{ENSL, CNRS, Laboratoire de physique, 69342 Lyon, France}

\author{\firstname{Yann} \lastname{Fraigneau}}
\address{Université Paris-Saclay, CNRS, Laboratoire Interdisciplinaire des Sciences du Numérique, 91405 Orsay, France}

\author{\firstname{Anne} \lastname{Sergent}}
\address[3]{Université Paris-Saclay, CNRS, Laboratoire Interdisciplinaire des Sciences du Numérique, 91405 Orsay, France}

\keywords{\kwd{Hydrodynamics} \kwd{Thermal Convection}\kwd{Turbulence}\kwd{Boundary layers}\kwd{Scaling laws}}
\altkeywords{\kwd{Hydrodynamique} \kwd{Convection thermique}\kwd{Turbulence}\kwd{Couches limites}\kwd{Lois d'échelle}}

\thanks{This project was provided with computing HPC and storage resources by GENCI at CINES and TGCC thanks to the grants 2022, 2023 and 2024-2A00326 on the supercomputer Joliot Curie’s ROME and Adastra’s GENOA partitions}
\CDRGrant[GENCI]{2022-2A00326}
\CDRGrant[GENCI]{2023-2A00326}
\CDRGrant[GENCI]{2024-2A00326}

\begin{abstract}
Most theoretical studies of Rayleigh--Bénard Convection assume that the velocity boundary layer develops on the whole width or height $H$ of the convection cell, or that the development length $h$ scales with $H$. We argue that it is probably not the case, and examine the consequences of an intermediate asymptotics hypothesis in the sense of Barenblatt~\cite{Barenblatt}, the development length $h$ scaling with the Nusselt number: $h\propto H\mathrm{Nu}^{-\alpha}$. This hypothesis is checked through existing experimental data, which show that $\alpha$ can take several different values. The analysis of Grossmann and Lohse~\cite{Lohse} is reexamined with this new point of view, stressing on pure scaling regimes.
\end{abstract}

\begin{altabstract}
La plupart des études théoriques sur la convection de Rayleigh--Bénard supposent que la couche limite de vitesse se développe sur toute la largeur ou la hauteur $H$ de la cellule de convection, ou que la longueur de développement $h$ est proportionnelle à $H$. Nous soutenons qu'il est probablement incorrect de faire cette supposition, et nous examinons les conséquences d'une hypothèse d'asymptotique intermédiaire au sens de Barenblatt~\cite{Barenblatt}, où la longueur de développement $h$ dépend du nombre de Nusselt: $h
\propto H\mathrm{Nu}^{-\alpha}$. Cette hypothèse est vérifiée à partir de données expérimentales existantes, qui montrent que $\alpha$ peut prendre plusieurs valeurs différentes. L'analyse de Grossmann et Lohse~\cite{Lohse} est réexaminée sous cette nouvelle perspective, en mettant l'accent sur les régimes d'échelle pure.
\end{altabstract}

\dateposted{2025-09-24}
\begin{document}

\maketitle

\section{Table of notations}

\begingroup
\renewcommand{\arraystretch}{0.7}
\setlength\extrarowheight{6pt}
\begin{tabular}{clc}
\textbf{Symbol} & \textbf{Definition} &
\begin{tabular}{@{}c@{}}
\small \textbf{1\textsuperscript{st} eq.\ of}\\[-1ex]
\small \textbf{occurrence}
\end{tabular}\\
\\[-2ex]
\hline\\[-2.7ex]
\hline\\[-2ex]
$u_i$ 		& $i$-component of the velocity &\eqref{NS} \\
$\theta$ 	& Local temperature fluctuation &\eqref{NS} \\
$\nu$ 		& Kinematic viscosity &\eqref{NS} \\
$\beta$ 		& Thermal expansion &\eqref{NS} \\
$\kappa$ 	& Heat diffusivity &\eqref{temp} \\
$g$ 			& Gravitational acceleration &\eqref{ra} \\
$H$ 			& Height of the cell &\eqref{ra} \\
$\Delta$ 	& Temperature difference between plates &\eqref{ra} \\
$\Rayleigh$	& Rayleigh number &\eqref{ra} \\
$\Prandtl$	& Prandtl number &\eqref{pr} \\
$L$ 			& Horizontal dimension of the cell &\eqref{gam} \\
$\Gamma$ 	& Aspect ratio &\eqref{gam} \\
$\dot{Q}$ 	& Vertical heat flux &\eqref{nu} \\
$C_p$ 		& Isobaric heat capacity of the fluid &\eqref{nu} \\
$\Nusselt$ 	& Nusselt number &\eqref{nu} \\
$U$ 			& Typical velocity of the fluid&\eqref{re} \\
$\Reynolds$	& Reynolds number &\eqref{re} \\
$\Theta$ 	& Typical temperature fluctuations&\eqref{ft} \\
$\fluct$  	& Relative temperature fluctuations &\eqref{ft} \\
$\mu$ 		& Exponent of the $\Nusselt$ \vs\ $\Rayleigh$ relation &\eqref{plnu} \\
$\varepsilon$	& Viscous dissipation per unit mass &\eqref{exactv} \\
$\epsilon_T$ 	& Thermal dissipation &\eqref{exactt} \\
$\chi$ 			& Vertical velocity-temperature correlation coefficient &\eqref{chi} \\
$h$ 				& Development length of the boundary layer	& \eqref{alp} \\
$\alpha$ 		& Exponent of $h$ \vs\ $\Nusselt$ dependence&\eqref{alp} \\
$\delta$ 		& Size of the thermal boundary layer &\eqref{ch} \\
$\gamma$ 		& Horizontal velocity vertical gradient &\eqref{ux} \\
$A$, $K$ 		& Constants of the Ching's solution &\eqref{correspondance} \\
$\Peclet$ 		& Peclet number &\eqref{pe+} \\
$\lambda$ 		& Size of the velocity boundary layer &\eqref{lambdel-} \\
$L_p$ 			& Total length of the plumes on the region of measurement &\eqref{Lp} \\
$S$ 				& Area of the region of measurement &\eqref{Lp} \\
$\varepsilon_{\bl}$	& Boundary layer contribution to the viscous dissipation &\eqref{epsBla} \\
$v_*$ 				& Friction velocity &\eqref{glog} \\
$y_+$				& Reduced distance to the plate &\eqref{y+} \\
$\DVB$ 				& Relative part of the bulk in the viscous dissipation &\eqref{dvb} \\
$\DVL_l$, $\DVL_u$ 	& Relative part of the boundary layer in the viscous dissipation &\eqref{dvll} \\
$\Delta^*$ 			& Twice the temperature difference across a thermal boundary layer &\eqref{bth} \\
$\sigma$ 			& Thermal dissipative scale in the bulk &\eqref{dtb} \\
$\vartheta$ 			& Temperature fluctuation at the thermal dissipative scale &\eqref{dtb} \\
$v_{\sigma}$ 		& Velocity fluctuation at the thermal dissipative scale &\eqref{dv} \\
$\eta$ 				& Viscous dissipative scale in the bulk &\eqref{dv2} \\
$v_{\eta}$ 			& Velocity fluctuation at the viscous dissipative scale &\eqref{dv2} \\
\end{tabular}
\endgroup

\section{Introduction}\label{int}

Convection is ubiquitous. Among other important problems, it governs the behavior of stars, the climate of planets, or the behavior of the molten core in a damaged nuclear plant. However, the bridge between these large scale events and their laboratory counter part, the Rayleigh--B\'enard convection cell is not yet firmly set.

The Rayleigh--B\'enard convection cell consists in a Newtonian, nearly incompressible, isotropic fluid, of kinematic viscosity $\nu$, of heat diffusivity $\kappa$, of thermal expansion $\beta$, put into motion by a vertical heat flux $\dot{Q}$. The problem is generally treated within the Boussinesq approximation, whose equations write:
\begin{align}
\partial_tu_i +u_j\partial_ju_i & = -\frac{1}{\rho}\partial_ip+g_i\beta\theta+\nu\partial_j\partial_ju_i, \label{NS}
\\
\partial_ju_j & = 0, \label{incomp}
\\
\partial_t\theta +u_j\partial_j\theta & = \kappa\partial_j\partial_j\theta, \label{temp}
\end{align}
where $g_i$ and $u_i$ are the $i$ components of the gravitational acceleration and the velocity, and $\theta$ is the temperature.

The non dimensional control parameters are the Rayleigh number:
\begin{equation}\label{ra}
\Rayleigh=\frac{g\beta \Delta H^3}{\nu\kappa},
\end{equation}
where $g$ is the gravitational acceleration, $\Delta$ the temperature difference between the horizontal plates, the hot one (generally below) and the cold one, $H$ the vertical distance between these plates. Of importance are also the Prandtl number:
\begin{equation}\label{pr}
\Prandtl=\frac{\nu}{\kappa},
\end{equation}
and the parameters specifying the shape of the cell, as the aspect ratio:
\begin{equation}\label{gam}
\Gamma=\frac{L}{H},
\end{equation}
where $L$ is the typical horizontal dimension of the plates.

The non dimensional parameters measuring the response of the cell are the Nusselt number:
\begin{equation}\label{nu}
\Nusselt=\frac{\dot{Q} H}{C_p\kappa \Delta},
\end{equation}
where $C_p$ is the isobaric heat capacity of the fluid per unit volume. The Reynolds number:
\begin{equation}\label{re}
\Reynolds=\frac{U H}{\nu},
\end{equation}
where $U$ is a typical velocity of the fluid, and the relative temperature fluctuations:
\begin{equation}\label{ft}
\fluct=\frac{\Theta}{\Delta},
\end{equation}
where $\Theta$ is a typical temperature fluctuation in the bulk, generally depend on the way and the point where the measures of $U$ and $\Theta$ are made, but are nevertheless precious indications of the nature of the flow.

During the second half of the past century, many propositions have been made for the $\Rayleigh$ dependence of $\Nusselt$, most of them as a power law (for a review, see~\cite{Lohse,Chilla,Ahlers,LS}):
\begin{equation}\label{plnu}
\Nusselt=\Rayleigh^{\mu},
\end{equation}
with occasional logarithmic corrections. Absolute majoration has been derived~\cite{Doering}, showing that, whatever the Prandtl number, in the limit of large $\Rayleigh$:
\begin{equation}\label{maj}
\Nusselt<C\Rayleigh^{1/2}, \quad \text{where $C$ is a constant}.
\end{equation}

Within approximately the same period, a great number of experiments~\cite{Lohse,Chilla,Ahlers} have been performed, including numerical experiments, reaching up to $\Rayleigh=10^{17}$, generally analyzed as power laws, equation~\eqref{plnu}, or a succession of power laws. It results in a wide variety of $\mu$ values proposed, both theoretically and experimentally.

In the past two decades, in a series of seminal papers, S.~Grossmann and D.~Lohse (GL) (see~\cite{GL}; for the last version, see~\cite{GL_last}) have proposed to rely the predictions to two exact results concerning the viscous ($\varepsilon$) and thermal ($\epsilon_T$) dissipations per unit mass:
\begin{align}
\varepsilon = \nu \langle \partial_iu_j\partial_iu_j \rangle & = \frac{\kappa^3}{H^4}\Rayleigh \, (\Nusselt-1) \, \Prandtl, \label{exactv}
\\
\epsilon_T = \kappa \langle \partial_iT\partial_iT\rangle & = \frac{\kappa \Delta^2}{H^2}\Nusselt.\label{exactt}
\end{align}
As remarked by Lohse and Shishkina~\cite{LS}, these relations are known since long, and can be traced at least to the works of Malkus~\cite{Malkus}.

Remarking that each dissipation can be dominated either by the boundary layer or by the bulk, they proposed to classify the flows into four categories:
%\begin{center}
%\begin{tabular}{cl}
%\hline
%		&\\*[-2.7ex]
%\hline
%(I)		& Both dissipations are dominated by the boundary layer \\
%\hline
%(II)		& Thermal dissipation dominated by the boundary layer, viscous by the bulk \\
%\hline
%(III) 	& Viscous dissipation dominated by the boundary layer, thermal by the bulk \\ 
%\hline 
%(IV) 	& Both dissipations are dominated by the bulk \\
%\hline
%		&\\*[-2.7ex]
%\hline
%\end{tabular}
%\end{center}
\begin{enumerate}\Romanenumi
\item \label{I} both dissipations are dominated by the boundary layer;
\item \label{II} thermal dissipation dominated by the boundary layer, viscous by the bulk;
\item \label{III} viscous dissipation dominated by the boundary layer, thermal by the bulk;
\item \label{IV} both dissipations are dominated by the bulk.
\end{enumerate}
Moreover, each category has the subscript ``u'' for the large $\Prandtl$, and ``l'' for the small ones, and has a prime ``$^\prime$'' if the boundary layer is turbulent. Evaluating the bulk and boundary layer dissipations, both viscous and thermal, and using the exact relations~\eqref{exactv} and~\eqref{exactt}, GL could obtain the $\Nusselt$ and $\Reynolds$ numbers in a wide range of intermediate values of $\Rayleigh$ and $\Prandtl$.  For instance, the bulk dissipation per unit mass is often evaluated as~\cite{Xav,GL}:
\begin{equation}\label{ebfull}
\varepsilon_b =\frac{U^3}{H}=\frac{U^3}{H^3}\frac{\nu^3}{\kappa^3}\frac{\kappa^3}{H^4}=\frac{\kappa^3}{H^4}\Reynolds^3\Prandtl^3.
\end{equation}
Assuming that the viscous dissipation is dominated by the bulk then gives, using equation~\eqref{exactv}:
\begin{equation}\label{eb}
\Rayleigh \, (\Nusselt-1) =\Reynolds^3\Prandtl^3.
\end{equation}
Ad hoc cross-over functions, and adjustable parameters were necessary, fitted to the existing experimental data. As a result, as remarked by GL, none of the pure regimes described in the above table appears alone, the combination of two of them being undistinguishable from a pure power law as equation~\eqref{plnu}.

In this paper, we shall follow the same way, relying to the exact results~\eqref{exactv} and~\eqref{exactt}, and evaluating the bulk and boundary layer dissipations. However, our approach will differ from the GL one on the following points:
\begin{itemize}
\item We shall take into account the various possible regimes, both in the bulk and in the boundary layer. We shall stress the influence of the so called mixing transition~\cite{Turb,Batchelor} in the bulk, and we shall reinterpret the mixing buffer zone of Kadanoff \etal~\cite{K}.
\item While recognizing the possibility of crossovers we shall stress on pure regimes, finding evidences of them in published experimental data.
\item We renounce to the dogma of single $\Nusselt$ or $\Reynolds$ values for given $\Rayleigh$ and $\Prandtl$, even at a given aspect ratio.
\end{itemize}

Indeed, such different regimes, at same $\Rayleigh$, $\Prandtl$, and $\Gamma$ numbers, have been observed not only between different experiments, as the Wu~\cite{Wu} and Chavanne~\cite{Xav} ones, but also within the same experiment~\cite{Cioni,RocheC,Rastello}. Mechanisms have been proposed for such multi-stability~\cite{Dubrulle}. We shall not treat this subject, focusing on the simple description of possible regimes.

The paper is organized as follows. In Section~\ref{cc}, we shall check a hypothesis made in several models, that the product of the typical velocity and the typical temperature fluctuations well represents their cross-correlation, \ie\ that their cross-correlation coefficient is constant. In Section~\ref{dl}, we shall introduce the development length of the velocity boundary layer, and make the hypothesis of a simple scaling of this length with the Nusselt number. We shall examine how it influences the velocity and thermal boundary layers. In Section~\ref{h}, we shall check the above hypothesis. In Section~\ref{disv}, we shall evaluate the viscous dissipation, in the boundary layer, distinguishing between Blasius or logarithmic ones, and in the bulk, distinguishing between soft and hard turbulence. This goes through a determination of the velocity boundary layer width, $\lambda$, which we use in Section~\ref{delt} for determining the thermal one $\delta$. In Section~\ref{dist}, we evaluate the thermal dissipation, particularly in the bulk, where we distinguish between soft and hard turbulence. In Section~\ref{dis}, we use all our results for discussing some remarkable cases, before to conclude in Section~\ref{conc}.

\section{The cross-correlation between temperature and vertical velocity}\label{cc}

We shall begin examining an assumption made by several models, the proportionality between the temperature and vertical velocity correlation $\langle u_z\theta\rangle $, and the product of the typical velocity $U$ with the typical temperature fluctuation $\Theta$. Noting that:
\begin{equation}\label{varnu}
\Nusselt=\frac{\langle u_z\theta\rangle H}{\kappa\Delta},
\end{equation}
the coefficient of cross-correlation can be obtained as:
\begin{equation}\label{chi}
\chi=\frac{\langle u_z\theta\rangle }{U\Theta}=\frac{\Nusselt}{\Reynolds \, \Prandtl}\frac{\Delta}{\Theta}.
\end{equation}

Unfortunately, very few experimental or numerical work give simultaneous estimations of $\Nusselt$, $\Reynolds$ and temperature fluctuations for given $\Rayleigh$ and $\Prandtl$ numbers. We had access to two of them: the PhD thesis of X.~Z.~Wu~\cite{Wu} and the work of Chavanne \etal~\cite{Xav}. For the latter, we used unpublished complementary materials, namely long simultaneous records of two close by thermometers, whose cross-correlation allowed to determine the Reynolds number.

Roche \etal~\cite{Roche1}, and independently Ahlers~\cite{Ahlers1}, realized that the influence of the walls is much larger than previously assumed. For both the above sets of data, we applied the correction proposed by Roche \etal~\cite{Roche1} to the Nusselt values, to take into account the finite heat conductivity of the walls. For more recent data, we assumed the corrections made.

We also used data from the work of Musilova \etal~\cite{Musilova}. We obtained an estimation of $\fluct$ (equation~\eqref{ft}) for these data using a fit of their results, shown on their Figure~6:
\begin{equation}\label{fitft}
\fluct_M=\parens[\big]{0.433+0.0125\times \log_{10}(\Rayleigh)} \, \Rayleigh^{-1/7}.
\end{equation}

\begin{figure}
\includegraphics[width= 8cm]{CorrelationSergentWu}
\caption{Open symbols: the \enquote{correlation coefficient} $\chi$ for the X.~Z.~Wu results~\cite{Wu}. Full symbols: the true correlation coefficient $\langle u_z\theta\rangle /\sqrt{\langle u_z^2\rangle \langle \theta^2\rangle }$ at various places for a numerical simulation of a rectangular Rayleigh--B\'enard cell of dimensions~$1$, $0.25$, $1$~\cite{Sergent1,Sergent2}. Black triangles: point $(0.25,0.125,0.5)$. Red squares: point $(0.35,0.125,0.5)$. Green diamonds: point $(0.5,0.125,0.5)$ (middle of the cell).\label{CorrW}
}
\end{figure}

The data of Wu~\cite{Wu}, Figure~\ref{CorrW}, show large variations for $\Rayleigh<10^8$, and an approximately constant value for larger $\Rayleigh$. We plot on the same figure the values of the local true correlation coefficient $\langle u_z\theta\rangle /\sqrt{\langle u_z^2\rangle \langle \theta^2\rangle }$ at various places for a numerical simulation of a rectangular Rayleigh--Bénard cell of dimensions~$1$, $0.25$, $1$~\cite{Sergent1,Sergent2}. The simulation again shows two regimes, one variable, and one constant. The apparent agreement of the numerical values for the simulation and the Wu experiment is probably fortuitous, as neither the shapes of the cells, nor the Prandtl numbers are the same. The succession of regimes in the simulation is more similar to the Musilova \etal~\cite{Musilova} one (see Figure~\ref{CorrX}). The remarkable point for the simulation is that the correlation in the center, where the fluctuations are very weak, is approximately the same as in much more active regions.

The data of Chavanne \etal~\cite{Xav}, Figure~\ref{CorrX}, show two plateaus, separated by a transition in the neighborhood of $\Rayleigh \simeq 3\cdot 10^9$. The data of Musilova \etal~\cite{Musilova} show large variations at low $\Rayleigh$, but are in close agreement with Chavanne \etal\ for $\Rayleigh>10^{10}$.

\begin{figure}[h!]
\includegraphics[width= 8cm]{CorrelationXavBrno}
\caption{The correlation coefficient $\chi$ for the Chavanne \etal\ results~\cite{Xav}, black diamonds, and the Musilova \etal\ ones~\cite{Musilova}, red circles. For the Chavanne results, we have estimated the uncertainty $s_1$ on the first plateau through $s_1^2={\parens[\big]{\sum_{i=1}^6{\parens[\big]{y_i-\langle y\rangle }}^2}}/5$ as we have 6 points. We operate similarly for the uncertainty $s_2$ on the second plateau, where we have 9 points. The difference between the two plateaus is five times $s=\sqrt{s_1^2+s_2^2}$.}\label{CorrX}
\end{figure}

We thus have different cases. In some ranges, the cross-correlation coefficient between velocity and temperature cannot be considered as constant. The basic assumption made in many models thus fails. On the other hand, however, there are ranges where the cross-correlation coefficient is constant. The usual approaches can then be used.

\section{The development length. Relation between thermal and velocity boundary layers}\label{dl}

In this section, our analysis will be at variance with most of previous ones, which consider that the velocity boundary layers develop on the whole width or height of the cell. We shall explicitly introduce the development length $h$ as a new parameter (see Figure~\ref{devlh}), and we shall assume that it is related to the Nusselt number as:
\begin{equation}\label{alp}
\frac{h}{H}=\Nusselt^{-\alpha}.
\end{equation}

In all this paper, we shall renounce to precisely determine the constant factors, so our equality rather means proportionality. The above relation is inspired by the idea that the thermal boundary layer in some way governs the velocity one.

\begin{figure}[h!]
\includegraphics[width= 10cm]{SchemaPlumes}
\caption{Sketch of a descending plume impacting the bottom plate, and two plumes rising, which shows our idea of what is the development length $h$.}\label{devlh}
\end{figure}

\subsection{Large Prandtl numbers}

To our knowledge, the only work which considers the possibility of a development length which differs from $H$ is that of E.~S.~C.~Ching~\cite{Ching}. Ching solved the problem of the vertical ($z$) temperature profile within the velocity boundary layer, for large Prandtl number. Under some conditions, she found a self similar solution:
\begin{equation}\label{ch}
T-T_p=\Delta f\parens[\big]{z/\delta(x)},
\end{equation}
$T_p$ being the temperature of the plate ($z=0$), and $x$ the velocity direction.

Rather than reproducing her argument, we propose to derive the typical value of $\delta$ from dimensional arguments. The $x$-component of the velocity has a linear profile:
\begin{equation}\label{ux}
u_x=\gamma z.
\end{equation}
Due to the strong anisotropy of the problem, we consider the dimensions $[x]$ and $[z]$ as independent. For example, $\delta$ is of dimension $[z]$, and $h$ is of dimension $[x]$. The basic dimensions are thus $[x]$, $[z]$ and the time $[t]$. $\delta$ depends on $\gamma$, the temperature diffusion coefficient $\kappa$, and the development length $h$. The dimensions are:
\begin{equation}\label{dim}
[u_x]=\frac{[x]}{[t]}, \quad [\gamma]=\frac{[x]}{[t][z]}, \quad [h]=[x], \quad [\kappa]=\frac{[z]^2}{[t]}, \quad [\delta]=[z].
\end{equation}
Thus, within a numerical factor:
\begin{equation}\label{delta}
\delta=\parens[\Bigg]{\frac{\kappa h}{\gamma}}^{1/3}.
\end{equation}
The solution of Ching depends on two parameters, $A$ and $K$, which are linked through $f(\infty)=1/2$. For small $K$, $A$ poorly depends on $K$, as $A=A_o(1-BK^2)$ with $B$ close to 1:
\begin{equation}\label{correspondance}
K=\delta^{\prime}\simeq \frac{\delta}{h}, A=\frac{\delta(\gamma \delta^2)^{\prime}}{2\kappa} \simeq \frac{\gamma\delta^3}{\kappa h}.
\end{equation}
``$^{\prime}$'' means deriving versus $x$, which is equivalent to divide by $h$ in our approach. We thus agree with the result of Ching, as far as $K$ is not too large, which means $h$ much larger than $\delta$.

Let us now remark that the above result, equation~\eqref{delta}, is equivalent to postulate that a peculiar Peclet number is 1 at the limit of the thermal boundary layer. Within the velocity boundary layer, we can determine the vertical velocity component $u_z$ through the mass conservation equation~\eqref{incomp}:
\begin{equation}\label{mass}
\dz u_z=-\dx u_x=\frac{\gamma}{h}z.
\end{equation}
Then $u_z(\delta)=\gamma\delta^2/h$ (we omit the factor 2), and:
\begin{equation}\label{pe+}
Pe=\frac{u_z\delta}{\kappa}=\frac{\gamma\delta^3}{h\kappa}=1.
\end{equation}

As often postulated (see for instance~\cite{Kraichnan}), the thermal boundary layer limit is the point where the Peclet number is 1. But we must remember that the velocity which enters this Peclet number is the vertical velocity.

A similar idea can be applied to the velocity boundary layer $\lambda$. At $z=\lambda$, $u_x$ reaches its main flow value. Using equation~\eqref{ux}:
\begin{equation}\label{lambda}
\lambda=\frac{U}{\gamma}.
\end{equation}
But, for a Blasius boundary layer, $\lambda$ is also the diffusion length of momentum during a time $h/U$:
\begin{equation}\label{lambda2}
\lambda=\sqrt{\frac{\nu h}{U}}.
\end{equation}
Using now equation~\eqref{mass}:
\begin{equation}\label{re1}
\frac{u_z(\lambda)\lambda}{\nu}=\frac{U\lambda^2}{h\nu}=1.
\end{equation}
The Reynolds number based on the vertical velocity is 1 at $z=\lambda$. The velocity boundary layer is the point where the momentum convective transport exceeds the diffusive one. The previous equation can also write:
\begin{equation}
\frac{\gamma\lambda^3}{h\nu}=\frac{\lambda^3}{\Prandtl\delta^3}=1,
\end{equation}
which gives the well known~\cite{Landau} relation between $\lambda$ and $\delta$ for large Prandtl numbers, and Blasius boundary layers:
\begin{equation}\label{lambdel+}
\lambda=\delta \Prandtl^{1/3}.
\end{equation}

\subsection{Low Prandtl numbers}

We can apply the above ideas to the case of small Prandtl numbers, where the thermal boundary layer extends out of the velocity one. Then, equation~\eqref{mass} writes:
\begin{equation}
\dz u_z=-\dx u_x=\frac{U}{h},
\end{equation}
which gives $u_z(\delta)=U\delta/h$, and:
\begin{equation}\label{pe-}
\Peclet=\frac{u_z\delta}{\kappa}=\frac{U\delta^2}{h\kappa}=1.
\end{equation}
At the velocity boundary layer limit, $\lambda$, we rather have $U\lambda^2/h\nu=1$. The consequence is that
\begin{equation}\label{lambdel-}
\lambda=\delta \Prandtl^{1/2},
\end{equation}
giving the relation between $\lambda$ and $\delta$ for small Prandtl numbers, and Blasius boundary layers.

\subsection{The transition between large and low Prandtl numbers}

The results of Ching allow to precisely determine the boundary between low and large Prandtl numbers. According to the Landau--Lifchitz textbook~\cite{Landau}, within the Blasius model, the velocity gradient at the boundary is:
\begin{equation}\label{gam2}
\gamma \simeq 0.332\sqrt{U^3/\nu x}=U/\lambda.
\end{equation}
Then, using equation~\eqref{correspondance}:
\begin{equation}\label{gam3}
2A\kappa=\delta{\parens{\gamma \delta^2}}^{\prime}=-\gamma^{\prime} \delta^3=\frac{\nu}{0.332^2}{\parens[\Bigg]{\frac{\delta}{\lambda}}}^3.
\end{equation}
In the small $K$ limit (where $h\gg\delta$), Ching obtains $A\simeq 2.136$:
\begin{equation}\label{prC}
{\parens[\Bigg]{\frac{\delta}{\lambda}}}^3=\frac{2A(0.332)^2}{\Prandtl}.
\end{equation}
The extrapolation to $\delta=\lambda$ gives $\Prandtl \simeq 0.471$ slightly larger than the estimation by Cioni \etal~\cite{Cioni} ($\Prandtl\simeq 0.2$) or Verzicco \etal~\cite{Verzicco1} ($\Prandtl\simeq 0.3$).

\section{A direct access to the development length}\label{h}

\subsection{By direct observation}

As many others, Puthenveettil \etal~\cite{PA} remarked that plumes generally detach from the plates as vertical sheets, thus appearing as lines in the plates observations.
 Indeed, these plumes and the corresponding lines can be observed by anyone looking at a cup of tea. Nice visualisation of them can be seen in Zocchi \etal~\cite{Zocchi}.
 But Puthenveettil \etal~\cite{PA} were the only ones, to our knowledge, to measure the length of these lines on a wide range of $\Rayleigh$ and $\Prandtl$. We agree with them that it is the thin ``local natural convection boundary layers'' around the plumes ``whose thickness essentially decides the heat flux in turbulent convection''. We thus shall assimilate the mean distance between the plumes with the development length $h$.

\begin{figure}
\includegraphics[width= 8cm]{devlengthPA}
\caption{$L_pH/S=H/h$ (see equation~\eqref{Lp}), where $h$ is the development length of the boundary layer for the Puthenveettil \etal\ results~\cite{PA}, versus the Nusselt number, in a logarithmic plot. It verifies equation~\eqref{alp} on a large range, with $\alpha \simeq 0.76$. Diamonds: $\Prandtl=0.7$; squares: $\Prandtl=6$; circles: $\Prandtl=602$.}\label{Lph}
\end{figure}

As Puthenveettil \etal~\cite{PA} remark, this mean distance is related to the total line length $L_p$ they measured and the area $S$ on which the measure is made through:
\begin{equation}\label{Lp}
h=\frac{S}{L_p}.
\end{equation}
We can thus check the relation~\eqref{alp}, looking at $L_pH/S$ versus $\Nusselt$. It is shown on Figure~\ref{Lph}. Except for the two smallest $\Nusselt$ values, the results nicely align as a power law. A linear regression on the data gives $\alpha \simeq 0.76$.\footnote{In their paper \cite{PA}, Puthenveettil \etal\ postulate that the Rayleigh number based on the distance between plumes, $\Rayleigh \, (h/H)^{3}$, only depends on $\Prandtl$. They write it $\Rayleigh^{1/3}h/H \propto \Prandtl^n$. Using equations~\eqref{alp}, \eqref{dev+}, and \eqref{Bla}, it gives $\alpha=0.8$ (very close to the $0.76$ found above) and $n=1/9$, close to the $0.1$ they find.}

\subsection{Through simultaneous \texorpdfstring{$\Nusselt$}{Nu} and \texorpdfstring{$\Reynolds$}{Re} measurements}

Equations~\eqref{pe+}, \eqref{pe-}, can be written in a different way, if the thermal dissipation is dominated by the boundary layer, thus in the GL cases~\eqref{I} and~\eqref{II}. Then $\Nusselt=H/\delta$ (we omit the constant fac\-tor~2), and, for $\Prandtl<1$:
\begin{equation}\label{dev-}
\frac{U\delta^2}{h\kappa}=\frac{UH}{\kappa}\frac{1}{\Nusselt^2}\frac{H}{h}=1 \spa \frac{H}{h}=\frac{\Nusselt^2}{\Reynolds \, \Prandtl},
\end{equation}
while, for $\Prandtl>1$, as $\gamma=U/\lambda$ and $\lambda/\delta=\Prandtl^{1/3}$:
\begin{equation}\label{dev+}
\frac{\gamma\delta^3}{h\kappa}=\frac{\delta UH}{\lambda\kappa}\frac{1}{\Nusselt^2}\frac{H}{h}=1 \spa \frac{H}{h}=\frac{\Nusselt^2}{\Reynolds \, \Prandtl^{2/3}}.
\end{equation}

Equations~\eqref{dev-}, \eqref{dev+}, give direct access to the behavior of $h$, independent of our assumption, equation~\eqref{alp}. Figures~\ref{devlW1} and~\ref{devlXav1} show how they apply to the results of respectively X.~Z.~Wu, Chavanne \etal\ and Musilova \etal\ The former show a power law dependence of $H/h$ versus $\Nusselt$, on a large final range, in agreement with the postulated equation~\eqref{alp}. A least squares regression for $\ln(\Nusselt)>3.3$ gives $\alpha \simeq 0.48$. The Chavanne \etal\ results show a transition between two such laws. A least squares regression for the lowest Nusselts gives $\alpha \simeq 0.30$. A least squares regression for $\ln(\Nusselt)>6$ gives $\alpha \simeq 0.77$, in agreement with the largest $\Nusselt$ values of the Musilova \etal\ results. The interest in Chavanne \etal\ set of data is the large span of Prandtl number values, which shows that no additional dependence with $\Prandtl$ exists out of the power law versus $\Nusselt$.

As noted above, the temperature-velocity correlation coefficient $\chi$ can be considered as constant, at least within a given regime. In the expressions of $H/h$, it allows to replace $\Nusselt/\Reynolds \, \Prandtl$ by $\Theta/\Delta$. Namely, for small Prandtl numbers:
\begin{equation}\label{dev-2}
\frac{H}{h}=\Nusselt\frac{\Theta}{\Delta},
\end{equation}
while, for large Prandtl numbers:
\begin{equation}\label{dev+2}
\frac{H}{h}=\Nusselt \, \Prandtl^{1/3}\frac{\Theta}{\Delta}.
\end{equation}
When the thermal dissipation is dominated by the boundary layer (cases~\eqref{I} and~\eqref{II} of GL), equations~\eqref{dev-2} and~\eqref{dev+2} have potentially another interpretation. Letting aside the Prandtl dependence, it can be written: $\delta\Delta=h\Theta$, as if the heat content of the thermal boundary layer would mix in a buffer layer of width $h$. This is close to the model proposed by Kadanoff \etal~\cite{K}. However, the Prandtl number dependence is hard to justify in this way.

\begin{figure}[h!]
\includegraphics[width= 8cm]{devlengthW1}
\caption{$\Nusselt^2/(\Reynolds \, \Prandtl^{2/3})=H/h$, where $h$ is the development length of the boundary layer for the X.~Z.~Wu results~\cite{Wu}, versus the Nusselt number, in a logarithmic plot. It verifies equation~\eqref{alp} on a large range, with $\alpha \simeq 0.48$.}\label{devlW1}
\end{figure}

\begin{figure}[h!]
\includegraphics[width= 8cm]{devlengthXavBrno}
\caption{$\Nusselt^2/(\Reynolds \, \Prandtl^{2/3})=H/h$, where $h$ is the development length of the boundary layer for the Chavanne \etal\ results~\cite{Xav}, asterisks, and the Musilova \etal\ ones~\cite{Musilova}, red circles, versus the Nusselt number, in a logarithmic plot. For Chavanne \etal, we have two regimes verifying equation~\eqref{alp}, with respective slopes $\alpha \simeq 0.30$ and $\alpha \simeq 0.77$. The final regime of Musilova \etal\ is similar to the Chavanne \etal\ one.}\label{devlXav1}
\end{figure}

\subsection{The \texorpdfstring{$\alpha$}{alpha}-spectrum}

We thus have an experimental confirmation of the conjecture made, equation~\eqref{alp}, but no explanation.  Up to this point, the exponent $\alpha$ has been an adjustable parameter. In this section we shall attempt to determine what are the possible $\alpha$ values, that is the $\alpha$ spectrum.

Indeed, the arguments presented by Kadanoff \etal~\cite{K} suggest that $\alpha$ should be equal to $0.5$. The following discussion mainly reproduce these arguments, which have been extended to low Prandtl numbers by Cioni \etal~\cite{Cioni}.

The simplest interpretation of the development length $h$, is that it represents the mean distance between plumes. Let us assume with Kadanoff \etal\ that sheet-like plumes, of thickness~$\lambda$ develop. Within such a plume, the two last terms of the Navier--Stokes equation~\eqref{NS} should equilibrate. For small Prandtl numbers, this gives:
\begin{equation}\label{K1-}
g\beta\Delta=\nu\frac{U}{\lambda^2}.
\end{equation}
This is equivalent to write that the buoyancy force $g\beta\Delta\lambda$, corresponding to the heat content~$\Delta\lambda$, equilibrates with the viscous force $\nu U/\lambda$. Equation~\eqref{K1-} gives:
\begin{equation}\label{K2-}
\frac{g\beta\Delta H^3}{\nu\kappa}=\Rayleigh= \frac{\nu}{\kappa}\frac{UH}{\nu}\frac{H^2}{\lambda^2}=\Reynolds\Nusselt^2,
\end{equation}
where we used the relation~\eqref{lambdel-} between $\lambda$ and $\delta$ for $\Prandtl<1$. Assuming that the bulk dominates the viscous dissipation (see equation~\eqref{eb}) and assimilating $\Nusselt$ and $(\Nusselt-1)$:
\begin{equation}\label{K3-}
\Rayleigh \, \Nusselt= \Reynolds\Nusselt^3=\Reynolds^3\Prandtl^2.
\end{equation}
Using equation~\eqref {dev-}:
\begin{equation}\label{K4-}
\Nusselt^3=\Reynolds^2\Prandtl^2=\Nusselt^{4-2\alpha}
\end{equation}
and $\alpha=0.5$.

The argument differs somewhat for $\Prandtl>1$. Then, $\delta<\lambda$, and the heat content of the plume, of thickness $\lambda$, is only $\Delta\delta$. Equation~\eqref{K1-} becomes:
\begin{equation}\label{K1+}
g\beta\Delta\delta=\nu\frac{U}{\lambda}.
\end{equation}
Then:
\begin{equation}\label{K2+}
\frac{g\beta\Delta H^3}{\nu\kappa}=\Rayleigh= \frac{\nu}{\kappa}\frac{UH}{\nu}\frac{H}{\lambda}\frac{H}{\delta}=\Prandtl^{2/3}\Reynolds\Nusselt^2,
\end{equation}
where we used the relation~\eqref{lambdel+} between $\lambda$ and $\delta$ for $\Prandtl>1$. Assuming again that the bulk dominates the viscous dissipation:
\begin{equation}\label{K3+}
\Rayleigh \, \Nusselt= \Prandtl^{2/3}\Reynolds\Nusselt^3=\Reynolds^3\Prandtl^2.
\end{equation}
Using equation~\eqref {dev+}:
\begin{equation}\label{K4+}
\Nusselt^3=\Reynolds^2\Prandtl^{4/3}=\Nusselt^{4-2\alpha}
\end{equation}
and $\alpha=0.5$. Moreover, in both cases, this model agrees with our assumption, equation~\eqref{alp}.

\begin{table}[h!]
\caption{Important relations for the development length $h$.}\label{t1}
\begin{tabular}{lcc}
\hline 		&&\\[-2.7ex]
\hline		&&\\[-2ex]
			& \textbf{Small $\mathrm{\mathbf{Pr}}$} & \textbf{Large $\mathrm{\mathbf{Pr}}$} \\
			&&\\[-2ex]
\hline 		&&\\[-2.7ex]
\hline		&&\\[-2ex]
Postulated	& $H/h=\Nusselt^{\alpha}$ 	& $H/h=\Nusselt^{\alpha}$ 		\\
			&&\\[-2ex]
\hline 		&&\\[-2ex]
Derived 		& $H/h=\Nusselt^2/\Reynolds \, \Prandtl$ 	& $H/h=\Nusselt^2/\Reynolds \, \Prandtl^{2/3}$ 	\\
			&&\\[-2ex]
\hline 		&&\\[-2ex]
Observed 	& 					& $H/h=L_pH/S$~\cite{PA}	\\
			&&\\[-2ex]
\hline 		&&\\[-2ex]
Observed $\alpha$ values & 		&
\begin{tabular}{@{}c@{}}
	$\alpha=0.3$; $\alpha=0.77$~\cite{Xav}
\\ 	$\alpha=0.48$~\cite{Wu}
\\	$\alpha=0.77$~\cite{PA} 
\end{tabular}	\\
			&&\\[-2ex]
\hline 		&&\\[-2ex]
Derived $\alpha$ values 	& $\alpha=0.5$~\cite{Cioni} 	&
\begin{tabular}{@{}c@{}}
	$\alpha=0.5$~\cite{K}
\\ 	$\alpha=0.5$; $\alpha=0.75$ (this work)
\end{tabular}	\\
			&&\\[-2ex]
\hline 		&&\\[-2.7ex]
\hline
\end{tabular}
\end{table}

The following argument could give an idea on the spectrum of other possible $\alpha$ values.

Let us call $\alpha_1$ the first possible value for $\alpha$. Above the transition from $\alpha=0$ to $\alpha=\alpha_1$, as $\Nusselt$ increases, and $\delta$ goes down, the development length $h$ could act as an effective large scale, the flow being confined between two neighboring plumes. Then, a new step of the instability of the thermal boundary layer could develop, yielding a new development length $h'$. The relation between $h$ and $h'$ would then write:
\begin{equation}\label{new}
\frac{h}{h'}={\parens[\Bigg]{\frac{h}{\delta}}}^{\alpha_1}
\end{equation}
or:
\begin{equation}\label{new2}
\frac{h}{H} \frac{H}{h'}=\Nusselt^{-\alpha_{n-1}} \frac{H}{h'}={\parens[\Bigg]{\frac{h}{H}\frac{H}{\delta}}}^{\alpha_1}={\parens{\Nusselt^{1-\alpha_{n-1}}}}^{\alpha_1}
\end{equation}
It gives the following recursion formula:
\begin{equation}\label{rec}
(1-\alpha_n)=(1-\alpha_{n-1})(1-\alpha_1).
\end{equation}

Identifying the Kadanoff $\alpha$ value, $0.5$, with $\alpha_1$, the following value would be $0.75$, very close to the observed ones, $0.77$ for the Chavanne \etal~\cite{Xav} or Musilova \etal~\cite{Musilova} data, or $0.76$ for the Puthenveettil \etal~\cite{PA} ones.

Note that for the lowest $\alpha$ value observed, $0.30$, the viscous dissipation is dominated by the boundary layer, in contradiction with the hypothesis of the Kadanoff \etal\ model~\cite{K,Cioni}.

\section{The viscous dissipation}\label{disv}

In this section we shall determine the viscous dissipation in the velocity boundary layer, and in the bulk. Using then equation~\eqref{exactv}, we shall identify $(\Nusselt-1)$ with $\Nusselt$, as the difference is smaller than the error bar in the experimental or numerical studies of the concerned regimes. A constant reference for the mean viscous dissipation per unit mass will be (see equation~\eqref{eb}):
\begin{equation}\label{ref}
\frac{U^3}{H}=\frac{\kappa^3}{H^4}\Reynolds^3\Prandtl^3.
\end{equation}

\subsection{The boundary layer contribution}\label{blcont}

We have here to distinguish between a Blasius type boundary layer and a logarithmic one.

\subsubsection{The Blasius case}

Then, $\lambda$, the width of the velocity boundary layer, is the momentum diffusion length during a time $h/U$:
\begin{align}
\lambda	& = \sqrt{\frac{\nu h}{U}}=H\Nusselt^{-\al/2}\Reynolds^{-1/2}, \label{gaBla1}
\\
\ga		& = \frac{U}{\lambda}=\frac{\nu}{H^2}\Nusselt^{\al/2}\Reynolds^{3/2}.\label{gaBla}
\end{align}
The boundary layer contribution to the viscous dissipation per unit mass will thus be:
\begin{equation}\label{epsBla}
\varepsilon_{\bl}=\frac{\lambda}{H}\nu\ga^2=\frac{\nu^3}{H^4}\Nusselt^{\al/2}\Reynolds^{5/2}.
\end{equation}
When this contribution dominates, comparing with equation~\eqref{exactv} gives:
\begin{equation}\label{Bla}
\Rayleigh \, \Nusselt=\Nusselt^{\al/2}\Reynolds^{5/2}\Prandtl^2.
\end{equation}
Another expression can be derived in this Blasius case. Namely:
\begin{equation}\label{epsBla2}
\varepsilon_{\bl}=\frac{\lambda}{H}\nu\frac{U^2}{\lambda^2}=\frac{\nu^3}{H^4}\Reynolds^2\frac{H}{\lambda}.
\end{equation}
Assuming that both the thermal and the viscous dissipations are dominated by the boundary layer, and using equations~\eqref{exactv}, \eqref{lambdel-} and~\eqref{lambdel+}, it gives, for small Prandtl numbers:
\begin{equation}\label{Bla-}
\Rayleigh \, \Nusselt=\Nusselt \, \Reynolds^2\Prandtl^{3/2},
\end{equation}
and for large Prandtl numbers:
\begin{equation}\label{Bla+}
\Rayleigh \, \Nusselt=\Nusselt \, \Reynolds^2\Prandtl^{5/3}.
\end{equation}
These expressions have the advantage to be independent of $\alpha$.

\subsubsection{The logarithmic case}

Again, we have to take into account the development length $h$. In the viscous sublayer, the horizontal velocity component is $u_x=\ga z$, with:
\begin{equation}
\ga=v_*^2/\nu,\label{glog}
\end{equation}
the transverse momentum flux being $\rho v_*^2$ by definition of $v_*$. As above, the vertical velocity component $u_z$ can be derived from the incompressibility relation~\eqref{incomp}:
\begin{equation}\label{uzlog}
\dz u_z =-\dx u_x=\frac{\ga z}{h} \spa u_z=\frac{\ga z^2}{h}
\end{equation}
(we omit the factor 2).

In the logarithmic range (see, for instance, \cite{Guyon}),
\begin{equation}
\dz u_x =\frac{v_*}{z} \spa u_x=v_*\ln\parens[\Bigg]{\frac{v_* z}{\nu}} \spa U=v_*\ln(\Reynolds)\label{uxlog}
\end{equation}
(we omit the von Karman constant).

The velocity sublayer limit $z=\lambda$ corresponds to the transition viscous-convective for the momentum flux. In the convective region, $\langle u_xu_z\rangle =v_*^2 \simeq \sqrt{\langle u_x^2\rangle \langle u_z^2\rangle }$, assuming again a constant correlation coefficient. In the viscous region, $\sqrt{\langle u_x^2\rangle }\simeq\ga z$, $\sqrt{\langle u_z^2\rangle }\simeq\ga z^2/h$. Thus, at the transition:
\begin{equation}\label{y+}
v_*^2 =\frac{\ga^2 \lambda^3}{h} \spa \frac{v_*^3 \lambda^3}{\nu^3}=y_+^3(\lambda)=\frac{v_*h}{\nu}=\frac{\Reynolds\Nusselt^{-\al}}{\ln(\Reynolds)}=D,
\end{equation}
$y_+=v_*z/\nu$ being the traditional notation for the reduced distance to the plate~\cite{Guyon}.

The above result strongly suggests that $D=1$. Indeed, to our knowledge, no experimental results show a dependence of the reduced distance $y_+$ at the top of the viscous sublayer with the Reynolds number. This is the option we shall take.

Then the contribution of the viscous sublayer to the viscous dissipation is:
\begin{equation}\label{disubl}
\frac{\lambda}{H}\nu\ga^2= \frac{v_*^3}{H}=\frac{U^3}{H(\ln \Reynolds)^3},
\end{equation}
while that of the logarithmic part is:
\begin{equation}\label{disubl2}
\frac{1}{H}\int_{\lambda}^H\langle u_xu_z\rangle \dz u_x \dd z= \frac{v_*^3}{H}\ln \Reynolds=\frac{U^3}{H(\ln \Reynolds)^2}.
\end{equation}
The latter being larger, we shall take it as the contribution of the logarithmic boundary layer. If it dominates the total viscous dissipation, we have:
\begin{equation}\label{log}
\Rayleigh \, \Nusselt=\frac{\Reynolds^3\Prandtl^2}{(\ln \Reynolds)^2}.
\end{equation}

\begin{rema}[On the relation between $\lambda$ and $\delta$]
Note that the above results have consequences both in the relation between the velocity boundary layer $\lambda$ and the thermal one $\delta$, and in the relation between the typical bulk temperature fluctuation $\Theta$ and $\Nusselt$ and $\Reynolds$.

As remarked above, our relations~\eqref{lambdel-} and~\eqref{lambdel+} are valid only if $h\gg\delta$. In the logarithmic case, the velocity vertical component fluctuation does not grow when $z>\lambda$, and remains of order $v_*$. For low Prandtl numbers in the logarithmic case, it implies:
\begin{equation}\label{dlowlog}
\delta=\frac{\lambda}{\Prandtl}.
\end{equation}
For $\Prandtl>1$ in the same case, let us remark that equation~\eqref{delta} implies, when $h \simeq \delta$:
\begin{equation}
\delta^2=\frac{\kappa}{\gamma}.
\end{equation}
Then, using equation~\eqref{glog}:
\begin{equation}\label{dlargeK}
\delta^2=\frac{\kappa\nu}{v_*^2}=\frac{\lambda^2}{\Prandtl}.
\end{equation}
Kraichnan~\cite{Kraichnan} obtained the same relation, assuming that the vertical velocity fluctuation is linear in $z$ for $z<\lambda$, at least for not too small $z$.

However, as Kraichnan himself remarked, such a regime cannot hold for large Prandtl numbers. The reason is that, as pointed above (see equation~\eqref{uzlog}), the vertical velocity must be quadratic in $z$ close to the plates. Using equation~\eqref{y+} :
\begin{equation}\label{dh}
\frac{\delta}{h}=\frac{\delta}{\lambda}\frac{\lambda}{h}=D^{-2/3}\Prandtl^{-1/2}.
\end{equation}
Thus the condition $h \simeq \delta$ cannot hold for large $\Prandtl$, where the relation between $\lambda$ and $\delta$ is, as in the Blasius case (see equation~\eqref{lambdel+}):
\begin{equation}\label{dlargepr}
\lambda=\delta \Prandtl^{1/3}.
\end{equation}
We shall call ``Kraichnan range'' the limited range of Prandtl numbers (if it even exists), close to 1, where equation~\eqref{dlargeK} holds.
\end{rema}

\begin{rema}[On the relation between $\Theta$ and $\Nusselt$, $\Reynolds$]
As for the relation between the typical bulk temperature fluctuation $\Theta$ and $\Nusselt$ and $\Reynolds$, as the velocity vertical component fluctuation does not grow when $z>\lambda$, and remains of order $v_*$, we should have:
\begin{equation}\label{fluxlog}
\langle u_z\theta\rangle =v_*\Theta.
\end{equation}
Then:
\begin{equation}\label{Nulog}
\Nusselt=\frac{\langle u_z\theta\rangle }{\kappa\Delta}=\frac{v_*H}{\nu}\frac{\nu}{\kappa}\frac{\Theta}{\Delta},
\end{equation}
and:
\begin{equation}\label{thetalog}
\frac{\Theta}{\Delta}=\frac{\Nusselt(\ln \Reynolds)}{\Reynolds \, \Prandtl},
\end{equation}
which substitutes to equation~\eqref{chi} with constant $\chi$ for logarithmic boundary layers.
\end{rema}

\subsection{The bulk contribution}

In the bulk, we shall follow the estimation by GL~\cite{GL} and determine the bulk contribution as $U^3/H$, both in the Blasius and in the logarithmic cases. It comes from the interaction between rising and falling plumes. If this contribution dominates, we have (see equation~\eqref{eb}):
\begin{equation}\label{bulk}
\Rayleigh \, \Nusselt=\Reynolds^3\Prandtl^2.
\end{equation}
We can now compare with the experimental results. Indeed, the quantity:
\begin{equation}\label{dvb}
\DVB=\frac{\Reynolds^3\Prandtl^2}{\Rayleigh \, \Nusselt}
\end{equation}
represents, within a constant factor, the part of the viscous dissipation which is due to the bulk, while the quantities:
\begin{equation}\label{dvll}
\DVL_l=\frac{\Reynolds^{2}\Prandtl^{3/2}}{\Rayleigh}
\end{equation}
(see equation~\eqref{Bla-}) for small Prandtl numbers, or:
\begin{equation}\label{dvlu}
\DVL_u=\frac{\Reynolds^{2}\Prandtl^{5/3}}{\Rayleigh}
\end{equation}
(see equation~\eqref{Bla+}) for large Prandtl numbers, represent, again within a constant factor, the part of the viscous dissipation which is due to the boundary layer. We let aside, for the moment, the logarithmic boundary layer case.

\begin{figure}[h!]
\includegraphics[width= 8cm]{dispviscW}
\caption{The respective parts of the viscous dissipation in the bulk ($\DVB$, squares), and in the boundary layer ($\DVL_u$, circles), for the X.~Z.~Wu results, using equations~\eqref{dvb} and~\eqref{dvlu}. A vertical shift has been applied for clarity.}\label{dispviscW}
\end{figure}

On Figure~\ref{dispviscW}, we show these two quantities, versus the Rayleigh number, in logarithmic scales, for the X.~Z.~Wu results. None of these quantities shows a plateau. The part of the bulk dissipation increases with $\Nusselt$, while the part of the boundary layer decreases. This is an excellent illustration of the GL point of view, a slow cross-over between the two regimes.

\begin{figure}[h!]
\includegraphics[width= 8cm]{dispviscXav}
\caption{The respective parts of the viscous dissipation in the bulk ($\DVB$, squares), and in the boundary layer ($\DVL_u$, circles), for the Chavanne \etal\ results, using equations~\eqref{dvb} and~\eqref{dvlu}. A vertical shift has been applied for clarity. Full symbols correspond to the first regime ($\alpha\simeq0.3$).}\label{dispviscXav}
\end{figure}

On Figure~\ref{dispviscXav}, we show the same quantities, versus the Rayleigh number, in logarithmic scales, for the Chavanne \etal\ results. While it is clear that the first regime, with $\al=0.30$, is dominated by the boundary layer, the second regime seems to give a plateau in both cases. We are unable to decide between the bulk or the boundary layer dominating the viscous dissipation.

\section{The thermal boundary layer \texorpdfstring{$\delta$}{delta}}\label{delt}

Indeed, the determination of $\lambda$ in the previous section gives the thermal boundary layer $\delta$ as a by-product. Let us first recall the results for $\lambda$. In the Blasius case:
\begin{equation}\label{lB}
\lambda=H\Nusselt^{-\alpha/2}\Reynolds^{-1/2},
\end{equation}
while, in the logarithmic boundary layer case:
\begin{equation}\label{llog}
\lambda=\frac{\nu}{v_*}=H\frac{\ln(\Reynolds)}{\Reynolds}.
\end{equation}
In the Blasius case, following our previous analysis, Section~\ref{blcont}, for small Prandtl numbers:
\begin{equation}\label{dlow}
\delta=\frac{\lambda}{\Prandtl^{1/2}},
\end{equation}
while, for large Prandtl numbers:
\begin{equation}\label{dlarge}
\delta=\frac{\lambda}{\Prandtl^{1/3}}.
\end{equation}
Reciprocally, in the logarithmic boundary layer case, for small Prandtl numbers:
\begin{equation}\label{dlow2}
\delta=\frac{\lambda}{\Prandtl},
\end{equation}
and, for large Prandtl numbers:
\begin{equation}\label{dlargelog}
\delta=\frac{\lambda}{\Prandtl^{1/3}},
\end{equation}
except for a small range of Prandtl numbers, close to 1 (the Kraichnan range), where (see equation~\eqref{dlargeK}):
\begin{equation}\label{dlarge2}
\delta=\frac{\lambda}{\Prandtl^{1/2}}.
\end{equation}

\begin{table}[ht!]
\caption{Relations between $\lambda$ and $\delta$.}\label{t2}
\begin{tabular}{lll}
\hline 		\\[-2.7ex]
\hline		\\[-2ex]
			& \textbf{Blasius boundary layer} & \textbf{Logarithmic boundary layer} \\
			\\[-2ex]
\hline 		\\[-2.7ex]
\hline		\\[-2ex]
Low $\Prandtl$& $\lambda=\delta \Prandtl^{1/2}=H\Nusselt^{-\alpha/2}\Reynolds^{-1/2}$ & $\lambda=\delta \Prandtl=H\Reynolds^{-1}\ln(\Reynolds)$ \\
			\\[-2ex]
Kraichnan range & $\lambda=\delta \Prandtl^{1/3}=H\Nusselt^{-\alpha/2}\Reynolds^{-1/2}$ & $\lambda=\delta \Prandtl^{1/2}=H\Reynolds^{-1}\ln(\Reynolds)$ \\
			\\[-2ex]
Large $\Prandtl$ &$\lambda=\delta \Prandtl^{1/3}=H\Nusselt^{-\alpha/2}\Reynolds^{-1/2}$ & $\lambda=\delta \Prandtl^{1/3}=H\Reynolds^{-1}\ln(\Reynolds)$ \\
			\\[-2ex]
\hline 		\\[-2.7ex]
\hline
\end{tabular}
\end{table}

\begin{table}[ht!]
\caption{Important relations for viscous dissipation. BL = boundary layer.}\label{t3}
\begin{tabular}{ccc}
\hline	\\[-2.7ex]
\hline	\\[-2ex]
					& \textbf{Flow's dominant part}\\
		\\[-2ex]
\hline 	\\[-2.7ex]
\hline	\\[-2ex]
Blasius BL  & Logarithmic BL & Bulk \\
		\\[-2ex]
\hline	\\[-2ex]
$\Rayleigh \, \Nusselt=\Nusselt^{\alpha/2}\Reynolds^{5/2}\Prandtl^2$ & $\Rayleigh \, \Nusselt=\Reynolds^3\Prandtl^2/\parens[\big]{\ln(\Reynolds)}^2$ &$\Rayleigh \, \Nusselt=\Reynolds^3\Prandtl^2$ \\
		\\[-2ex]
\hline 	\\[-2.7ex]
\hline
\end{tabular}
\end{table}

\section{The thermal dissipation}\label{dist}

\subsection{The boundary layer contribution}

The boundary layer contribution is simply given by:
\begin{equation}\label{thbl}
\kappa\parens[\Bigg]{\frac{\Delta}{\delta}}^2\frac{\delta}{H}=\kappa\parens[\Bigg]{\frac{\Delta}{H}}^2\frac{H}{\delta}.
\end{equation}
When the thermal dissipation is dominated by the boundary layer, equation~\eqref{exactt} yields an identity. We could refer to our determination of the boundary layer $\delta$, for instance through equations~\eqref{dev-} or~\eqref{dev+}, but we must remember that we used it to determine $\al$, and thus it is identically verified.

The only check we have is thus to look at the bulk contribution. If the bulk contribution is constant, we shall conclude that it dominates the thermal dissipation. If it is not constant, we shall conclude that a significant boundary layer contribution exists. We have no possible cross-check.

\subsubsection{The temperature profile}

Moreover, considering that the boundary layer contribution is only part of the total thermal dissipation implies that the temperature difference across the boundary layer, $\Delta^*/2$, differs from $\Delta/2$. Let us call $\Nusselt^*$ the boundary layer contribution to the thermal dissipation normalized by $\kappa(\Delta/H)^2$:
\begin{equation}\label{bth}
\frac{\kappa}{4}{\parens[\Bigg]{\frac{\Delta^*}{\delta}}}^2\frac{2\delta}{H}=\frac{\kappa\Delta^2}{H^2}{\parens[\Bigg]{\frac{\Delta^*}{\Delta}}}^2\frac{H}{2\delta}=\frac{\kappa\Delta^2}{H^2}\Nusselt^*.
\end{equation}
As:
\begin{equation}\label{cond1}
\frac{\Delta^*}{2}=\delta\frac{\dot{Q}}{\kappa C_p}=\frac{\delta}{H}\Delta \Nusselt,
\end{equation}
we have:
\begin{equation}\label{cond}
\frac{\Delta^*}{\Delta}=\frac{2\delta}{H}\Nusselt
\end{equation}
and:
\begin{equation}\label{nust}
\Nusselt^*=\frac{\Delta^*}{\Delta}\Nusselt,
\end{equation}
which means that the boundary layer contribution is in the same proportion as the temperature differences. It is then interesting to look at the temperature profile in the convective part of the convection cell.

The situation is indeed very similar to that yielding to the turbulent logarithmic velocity profile. $\Th=\Delta \Nusselt/\Reynolds \, \Prandtl$ acts as a typical temperature difference, while, at a distance $z$ from the plate, there is no other characteristic length than $z$. From dimensional analysis, the mean vertical temperature gradient must write:
\begin{equation}\label{grad}
\angles[\Bigg]{\frac{\partial T}{\partial z}} = \varpi\frac{\Th}{z},
\end{equation}
where $\varpi$ is a universal constant, similar to the von Karman constant. However, our arguments are independent of the velocity boundary layer being turbulent or of the Blasius type.

Then $\langle T\rangle =\varpi\Th\ln(z)+\cst$ and $\Delta -\Delta^*\simeq \varpi\Th\ln(\Nusselt)$. Note that such a logarithmic temperature profile has been effectively observed in some studies~\cite{log}.

Thus, the evolution is slow, and $\Delta$ cannot be much larger than $\Delta^*$ before extremely large values of $\Nusselt$ be reached. In practical situations, the boundary layer contribution is always a noticeable part of the total thermal dissipation. Rather than a dissipation ``dominated'' by the bulk, we should speak of a dissipation ``driven'' by the bulk, as the boundary layer adapts to follow the bulk contribution.

\subsection{The bulk contribution}

Following the approach of~\cite{Batchelor}, we shall determine the bulk contribution to the thermal dissipation through:
\begin{equation}\label{dtb}
\kappa \langle \di T\di T\rangle  \simeq \kappa\parens[\Bigg]{\frac{\vartheta}{\dv}}^2,
\end{equation}
where $\vartheta$ is the temperature fluctuation at the thermal dissipative scale $\dv$. As in~\cite{Batchelor}, we have to distinguish between hard and soft turbulence, and large or small Prandtl numbers.

\subsubsection{Small Prandtl numbers. Hard turbulence}

When the inertial turbulent cascade is sufficiently developed, the thermal dissipative scale $\dv$, while larger than the viscous one $\eta$ ($\Prandtl<1$), is in the inertial range: $H>\dv>\eta$. At the scale $\dv$, the thermal diffusion time is equal to the stretching time. Put in other words, the Peclet number is 1:
\begin{equation}\label{dv}
\frac{\kappa}{\dv^2}=\frac{v_{\dv}}{\dv}.
\end{equation}
Using the Kolmogorov 41 theory~\cite{Frisch}:
\begin{equation}\label{dv2}
\frac{H}{\dv}=\frac{H}{\kappa}U{\parens[\Bigg]{\frac{H}{\dv}}}^{1/3} \spa \frac{H}{\dv}={\parens[\Bigg]{\frac{UH}{\kappa}}}^{3/4}=(\Reynolds \, \Prandtl)^{3/4}.
\end{equation}
As for the temperature fluctuations:
\begin{equation}\label{dt}
\vt=\Th{\parens[\Bigg]{\frac{\dv}{H}}}^{1/3}=\Th(\Reynolds \, \Prandtl)^{-1/4}.
\end{equation}
Using equations~\eqref{dtb} and~\eqref{chi}, the bulk contribution to the thermal dissipation is then:
\begin{equation}\label{dtb1}
\kappa{\parens[\Bigg]{\frac{\Delta}{H}}}^2{\parens[\Bigg]{\frac{\Th}{\Delta}}}^2(\Reynolds \, \Prandtl)^{-1/2}(\Reynolds \, \Prandtl)^{3/2} =\kappa{\parens[\Bigg]{\frac{\Delta}{H}}}^2\frac{\Nusselt^2}{\Reynolds \, \Prandtl}.
\end{equation}
Note that we could have obtained this result, identifying the thermal dissipation $\epsilon_T$ with $U\Th^2/H$, as GL did~\cite{GL}. $U\Th^2/H$ is the large scale expression of $\epsilon_T$, and the cascade ensures the conservation of $\epsilon_T$:
\begin{equation}\label{dtbis}
\frac{U\Th^2}{H}=\kappa{\parens[\Bigg]{\frac{\Delta}{H}}}^2\frac{UH}{\nu}\frac{\nu}{\kappa}\frac{\Th^2}{\Delta^2}.
\end{equation}
If this contribution dominates, using equation~\eqref{exactt} gives:
\begin{equation}\label{nure1}
\Nusselt=\Reynolds \, \Prandtl \spa \frac{\Th}{\Delta}=\cst.
\end{equation}
If the bulk viscous dissipation also dominates, using equation~\eqref{bulk} gives:
\begin{equation}\label{ult1}
\Reynolds= \Rayleigh^{1/2}\Prandtl^{-1/2} \spa \Nusselt=\Rayleigh^{1/2}\Prandtl^{1/2}.
\end{equation}

\subsubsection{Very small Prandtl numbers. Hard turbulence}\label{vsp}

We consider this case, inspired by the recent paper of Shishkina and Lohse~\cite{SL}. These authors correctly remark that, for $\Prandtl<\Rayleigh^{-1}$, equation~\eqref{ult1} would imply a Nusselt number going to zero. Indeed, $\Nusselt$ doesn't go to zero. When it goes close to one, we must go back to equation~\eqref{exactv}. If the bulk viscous dissipation dominates, equation~\eqref{bulk} becomes:
\begin{equation}\label{bulk2}
\Rayleigh \, (\Nusselt-1)=\Reynolds^3\Prandtl^2.
\end{equation}
As:
\begin{equation}
\dot{Q}=C_p\parens[\Bigg]{\kappa\frac{\Delta}{H}+\langle u_z\theta\rangle},
\end{equation}
we can now estimate $\Theta$ through:
\begin{equation}\label{th2}
\frac{\Theta}{\Delta}=\frac{\Nusselt-1}{\Reynolds \, \Prandtl}.
\end{equation}
Equations~\eqref{exactt}, \eqref{dtbis}, \eqref{th2} give:
\begin{equation}\label{dtter}
\frac{U\Th^2}{H}=\kappa{\parens[\Bigg]{\frac{\Delta}{H}}}^2\Reynolds \, \Prandtl \, {\parens[\Bigg]{\frac{\Nusselt-1}{\Reynolds \, \Prandtl}}}^2=\kappa{\parens[\Bigg]{\frac{\Delta}{H}}}^2\Nusselt.
\end{equation}
Using equations~\eqref{bulk2}, \eqref{dtter} and $\Nusselt \simeq 1$, $(\Nusselt-1)=(\Reynolds \, \Prandtl)^{1/2}$, and:
\begin{equation}\label{ult2}
\Reynolds= \Rayleigh^{2/5}\Prandtl^{-3/5} \spa \Nusselt-1=\Rayleigh^{1/5}\Prandtl^{1/5}.
\end{equation}

\subsubsection{Small Prandtl numbers. Intermediate turbulence}

For smaller Reynolds numbers, the thermal dissipative scale $\dv$ is out of the inertial range. Then, we must identify it with $H$. The bulk contribution to the thermal dissipation is then:
\begin{equation}\label{dtb2}
\kappa{\parens[\Bigg]{\frac{\Th}{H}}}^2=\kappa{\parens[\Bigg]{\frac{\Delta}{H}}}^2{\parens[\Bigg]{\frac{\Th}{\Delta}}}^2=\kappa{\parens[\Bigg]{\frac{\Delta}{H}}}^2{\parens[\Bigg]{\frac{\Nusselt}{\Reynolds \, \Prandtl}}}^2.
\end{equation}
If this contribution dominates, using equation~\eqref{exactt} gives:
\begin{equation}\label{nure3}
\Nusselt=(\Reynolds \, \Prandtl)^2 \spa \frac{\Th}{\Delta}=\Reynolds \, \Prandtl.
\end{equation}
If the bulk viscous dissipation also dominates, using equation~\eqref{bulk} gives:
\begin{equation}\label{nure2}
\Nusselt=(\Reynolds \, \Prandtl)^2=(\Rayleigh \, \Prandtl)^2 \spa \frac{\Th}{\Delta}=\Reynolds \, \Prandtl.
\end{equation}

\subsubsection{Arbitrary Prandtl numbers. Soft turbulence}

In this case, while turbulent, the bulk has no inertial energy cascade. As in~\cite{Batchelor}, we then determine the temperature dissipation scale $\sigma$, writing that the inverse temperature diffusion time is equal to the velocity gradient, itself determined through the dissipation:
\begin{equation}\label{sig2}
\frac{\kappa}{\sigma^2}=\sqrt{\frac{\varepsilon}{\nu}}.
\end{equation}
The bulk contribution to the thermal dissipation is then, both for small and large Prandtl numbers:
\begin{equation}\label{dt3hpr}
\begin{split}
\kappa\frac{\langle \theta^2\rangle }{\sigma^2}&=\langle \theta^2\rangle \sqrt{\frac{\varepsilon}{\nu}}\\
&=\Delta^2\frac{\Nusselt^2}{\Reynolds^2\Prandtl^2}\sqrt{\frac{\kappa^3\Reynolds^3\Prandtl^3}{H^4\nu}}\\
&=\kappa\parens[\Bigg]{\frac{\Delta}{H}}^2\frac{\Nusselt^2}{\Reynolds^{1/2}\Prandtl}.
\end{split}
\end{equation}
If this contribution dominates, we have, for arbitrary Prandtl numbers:
\begin{equation}\label{nure5-}
\Nusselt=\Reynolds^{1/2}\Prandtl \spa \frac{\Th}{\Delta}=\Prandtl \, \Nusselt^{-1}.
\end{equation}
If the bulk viscous dissipation also dominates (GL case~\eqref{IV}), equations~\eqref{nure5-} and~\eqref{bulk} give:
\begin{equation}\label{nure6-}
\Reynolds=\Rayleigh^{2/5}\Prandtl^{-2/5} \spa \Nusselt=\Rayleigh^{1/5}\Prandtl^{4/5}.
\end{equation}

\subsubsection{Large Prandtl numbers. Hard turbulence}\label{thermalhpr}

In this case, an inertial range exists, but the thermal dissipative scale $\dv$ is smaller than the viscous dissipation scale $\eta$. These scales are related through:
\begin{equation}\label{dv2+}
\frac{\kappa}{\dv^2}=\frac{v_{\dv}}{\dv}=\frac{v_{\eta}}{\eta}=\frac{\nu}{\eta^2}.
\end{equation}
This regime is known as the Batchelor regime~\cite{Batchelor}. The temperature fluctuation at the thermal dissipative scale $\dv$ is the same as at the viscous dissipative scale $\eta$:
\begin{equation}\label{dt2}
\vt=\Th{\parens[\Bigg]{\frac{\eta}{H}}}^{1/3}=\Th \Reynolds^{-1/4}.
\end{equation}
The bulk contribution to the thermal dissipation is then:
\begin{equation}\label{dtb3}
\begin{split}
\kappa{\parens[\Bigg]{\frac{\vt}{\dv}}}^2&=\kappa{\parens[\Bigg]{\frac{\Delta}{H}}}^2{\parens[\Bigg]{\frac{\Th}{\Delta}}}^2{\parens[\Bigg]{\frac{\vt}{\Th}}}^2{\parens[\Bigg]{\frac{H}{\eta}}}^2{\parens[\Bigg]{\frac{\eta}{\dv}}}^2 \\
&=\kappa{\parens[\Bigg]{\frac{\Delta}{H}}}^2{\parens[\Bigg]{\frac{\Th}{\Delta}}}^2\Reynolds \, \Prandtl \\
&=\kappa{\parens[\Bigg]{\frac{\Delta}{H}}}^2\frac{\Nusselt^2}{\Reynolds \, \Prandtl}.
\end{split}
\end{equation}
This result is identical to what we obtained in the low Prandtl numbers case, equation~\eqref{dtbis}, for the same reasons.
 We shall show below that, as a consequence of the recently derived bounds by Choffrut \etal~\cite{Choffrut}, such a bulk dissipation cannot exceed the boundary layer thermal dissipation for arbitrarily large Rayleigh numbers.

Choffrut \etal\ derived two bounds for $\Nusselt$ in the large $\Prandtl$ case:
\begin{equation}\label{ch1}
\Nusselt<\Rayleigh^{1/2}\Prandtl^{-1/2},
\end{equation}
for $\Prandtl<\Rayleigh^{1/3}$, and:
\begin{equation}\label{ch2}
\Nusselt<\Rayleigh^{1/3},
\end{equation}
for $\Prandtl>\Rayleigh^{1/3}$.

Assume that the bulk thermal dissipation, equation~\eqref{dtb3}, be larger than the boundary layer one, equation~\eqref{thbl}. It would mean, for large Reynolds numbers:
\begin{equation}\label{ineq1}
\begin{aligned}
\frac{\Nusselt^2}{\Reynolds \, \Prandtl} &> \frac{H}{\delta} = \Reynolds \, \Prandtl^{1/3}, \\
\Nusselt^2&>\Reynolds^2\Prandtl^{4/3}.
\end{aligned}
\end{equation}
Using equation~\eqref{bulk}:
\begin{equation}\label{ineq2}
\begin{aligned}
\Rayleigh \, \Nusselt=\Reynolds^3\Prandtl^2 &> \Rayleigh \, \Reynolds \, \Prandtl^{2/3}, \\
\Reynolds^2\Prandtl^{4/3}&>\Rayleigh.
\end{aligned}
\end{equation}
Using equation~\eqref{ch1} ($\Prandtl<\Rayleigh^{1/3}$):
\begin{equation}\label{ineq3}
\Reynolds^2\Prandtl^{1/3}>\Rayleigh \, \Prandtl^{-1}>\Nusselt^2>\Reynolds^2\Prandtl^{4/3},
\end{equation}
which is impossible at large $\Prandtl$.

Using equation~\eqref{ch2} ($\Prandtl>\Rayleigh^{1/3}$) and equation~\eqref{ineq1}:
\begin{equation}\label{ineq5}
\begin{aligned}
\Rayleigh &> \Nusselt^3>\Reynolds^3\Prandtl^{2} =\Rayleigh \, \Nusselt, \\
\Nusselt^2&>\Rayleigh,
\end{aligned}
\end{equation}
which violates equation~\eqref{ch2}.

Thus, in each case, the thermal dissipation is dominated by the boundary layer.

\begin{table}[ht!]
\small
\caption{Important relations for thermal dissipation. BL = boundary layer.\label{t4}
}
\begin{tabular}{llll}
\hline	\\[-2.7ex]
\hline	\\[-2ex]
\textbf{Flow's dominant part} 	&\textbf{Low Prandtl numbers} & \textbf{Kraichnan range} & \textbf{Large Prandtl numbers} \\
		\\[-2ex]
\hline	\\[-2.7ex]
\hline	\\[-2ex]
Blasius BL		& $\Nusselt=\Nusselt^{\alpha/2}\Reynolds^{1/2} \Prandtl^{1/2}$ & $\Nusselt=\Nusselt^{\alpha/2}\Reynolds^{1/2} \Prandtl^{1/2}$ & $\Nusselt=\Nusselt^{\alpha/2}\Reynolds^{1/2} \Prandtl^{1/3}$ \\
								&&&\\[-2ex]
Logarithmic BL 					& $\Nusselt=\Reynolds \, \Prandtl/\ln(\Reynolds) $ & $\Nusselt=\Reynolds \, \Prandtl^{1/2}/\ln(\Reynolds) $ &$\Nusselt=\Reynolds \, \Prandtl^{1/3}/\ln(\Reynolds) $ \\
								&&&\\[-2ex]
Bulk Soft turb					& $\Nusselt=\Nusselt^2\Reynolds^{-1/2}\Prandtl^{-1/2} $ & $\Nusselt=\Nusselt^2\Reynolds^{-1/2}\Prandtl^{-1/2} $ &$\Nusselt=\Nusselt^2\Reynolds^{-1/2}\Prandtl^{-1/2} $ \\
								&&&\\[-2ex]
Bulk Intermediate turb 			& $\Nusselt=\Nusselt^2\Reynolds^{-2}\Prandtl^{-2}$ & Not dominant &Not dominant \\
								&&&\\[-2ex]
Bulk Hard turb					& $\Nusselt=\Nusselt^2\Reynolds^{-1}\Prandtl^{-1} $ & Not dominant & Not dominant \\
		\\[-2ex]
\hline 	\\[-2.7ex]
\hline
\end{tabular}
\end{table}

\section{Discussion}\label{dis}

We can now obtain the scalings of $\Nusselt$, $\Reynolds$, and $\Th/\Delta$ in all
\enquote{pure} regimes. Such an enumeration would however be tedious, as we have to distinguish not only all the GL regimes, but also if the Prandtl number is large or small, if the boundary layer is Blasius or logarithmic, and if the turbulence in the bulk is hard or soft. We prefer to detail some remarkable cases, to show how these scalings can be obtained.

\subsection{The Kadanoff solution}

This case corresponds to a bulk dominated viscous dissipation, and a thermal dissipation dominated by a Blasius boundary layer. The value of $\alpha$ is $\alpha=1/2$.

Equation~\eqref{bulk} gives:
\begin{equation}
\Rayleigh \, \Nusselt=\Reynolds^3\Prandtl^2.
\end{equation}
At large Prandtl number, equations~\eqref{lB} and~\eqref{dlarge} give:
\begin{equation}\label{Bl}
\begin{aligned}
\Nusselt&=\frac{H}{\delta}=\Nusselt^{\alpha/2}\Reynolds^{1/2}\Prandtl^{1/3}, \\
\Nusselt&=\Reynolds^{1/(2-\alpha)}\Prandtl^{2/3(2-\alpha)}.
\end{aligned}
\end{equation}
Then:
\begin{align}
\Reynolds&=\Rayleigh^{(2-\alpha)/(5-3\alpha)}\Prandtl^{-2/3}, \\
\Nusselt&=\Rayleigh^{1/(5-3\alpha)}, \\
\frac{\Th}{\Delta}&=\Rayleigh^{(\alpha-1)/(5-3\alpha)}\Prandtl^{-1/3},
\end{align}
which, for $\alpha=1/2$, gives:
\begin{equation}
\Reynolds=\Rayleigh^{3/7}\Prandtl^{-2/3} \spac \Nusselt=\Rayleigh^{2/7} \spac \frac{\Th}{\Delta}=\Rayleigh^{-1/7}\Prandtl^{-1/3}.
\end{equation}
The Cioni \etal~\cite{Cioni} extension of this regime to small Prandtl numbers corresponds to using equation~\eqref{dlow} instead of~\eqref{dlarge}:
\begin{equation}\label{Bs}
\begin{aligned}
\Nusselt&=\frac{H}{\delta}=\Nusselt^{\alpha/2}\Reynolds^{1/2}\Prandtl^{1/2}, \\
\Nusselt&=\Reynolds^{1/(2-\alpha)}\Prandtl^{1/(2-\alpha)}.
\end{aligned}
\end{equation}
Then:
\begin{align}
\Reynolds&=\Rayleigh^{(2-\alpha)/(5-3\alpha)}\Prandtl^{-(3-2\alpha)/(5-3\alpha)}, \\
\Nusselt&=\Rayleigh^{1/(5-3\alpha)}\Prandtl^{1/(5-3\alpha)}, \\
\frac{\Th}{\Delta}&=\Rayleigh^{(\alpha-1)/(5-3\alpha)}\Prandtl^{(5\alpha-1)/(5-3\alpha)},
\end{align}
which, for $\alpha=1/2$, gives:
\begin{equation}
\Reynolds=\Rayleigh^{3/7}\Prandtl^{-4/7} \spac \Nusselt=\Rayleigh^{2/7}\Prandtl^{2/7} \spac \frac{\Th}{\Delta}=\Rayleigh^{-1/7}\Prandtl^{3/7}.
\end{equation}
While the final regime of Wu results gives $\alpha=1/2$, we cannot identify this regime with the Kadanoff one. The reason is that the bulk relative part of the viscous dissipation is not constant, as can be seen on Figure~\ref{dispviscW}, which shows that the boundary layer part is significant. For a boundary layer part dominant, instead of equation~\eqref{bulk}, we should have (see equation~\eqref{Bla}):
\begin{equation}
\Rayleigh \, \Nusselt=\Nusselt^{\al/2}\Reynolds^{5/2}\Prandtl^2.
\end{equation}
Together with equations~\eqref{lB} and~\eqref{dlarge}, it gives:
\begin{equation}
\Reynolds=\Rayleigh^{1/2}\Prandtl^{-5/6},
\end{equation}
and, using $\alpha=1/2$:
\begin{equation}
\Nusselt=\Rayleigh^{1/3}\Prandtl^{-1/9}.
\end{equation}
As GL proposed~\cite{GL}, this final regime of Wu results presents a crossover between the above regime and the Kadanoff one.

In their paper, Castaing \etal\ compared $\Theta/\Delta$ and Nusselt. The two above equations give:
\begin{equation}
\frac{\Theta}{\Delta}=\frac{\Nusselt}{\Reynolds \, \Prandtl} =\Nusselt^{-1/2}\Prandtl^{-7/18},
\end{equation}
while the Kadanoff solution gives:
\begin{equation}
\frac{\Theta}{\Delta}=\frac{\Nusselt}{\Reynolds \, \Prandtl} =\Nusselt^{-1/2}\Prandtl^{-1/3}.
\end{equation}
The difference is too small to conclude.

\subsection{The Kraichnan solution}

In a celebrated paper~\cite{Kraichnan}, R.~Kraichnan proposed a large Rayleigh numbers regime for moderately large Prandtl numbers, which has long been understood as the ``ultimate'' regime. Each thermal boundary layer is supposed to have half the total temperature difference $\Delta$ across~it, in agreement with the conclusions of Section~\ref{thermalhpr}, and the Choffrut \etal~\cite{Choffrut} bounds.

On the one hand, we again have, using equation~\eqref{bulk}:
\begin{equation}
\Rayleigh \, \Nusselt=\Reynolds^3\Prandtl^2.
\end{equation}

On the other hand, using equations~\eqref{dlargeK} and~\eqref{llog} we obtain:
\begin{equation}\label{K2}
\Nusselt=\frac{H}{\delta}=\frac{\Reynolds \, \Prandtl^{1/2}}{\ln(\Reynolds)}.
\end{equation}
It gives:
\begin{align}
\Reynolds&=\frac{\Rayleigh^{1/2}\Prandtl^{-3/4}}{\parens[\big]{\ln(\Reynolds)}^{1/2}}, \\
\Nusselt&=\frac{\Rayleigh^{1/2}\Prandtl^{-1/4}}{\parens[\big]{\ln(\Reynolds)}^{3/2}},
\end{align}
which is the result of Kraichnan. Using equation~\eqref{thetalog}:
\begin{equation}
\frac{\Th}{\Delta}=\Prandtl^{-1/2}.
\end{equation}

\subsection{The observation of Cioni \texorpdfstring{\etal}{et al.}}

During one of the runs of their mercury experiment, Cioni \etal~\cite{Cioni} observed a rapid increase of the Nusselt number on a limited range of Rayleigh numbers. Their interpretation was a transition between two regimes. However, as shown on the Figure~\ref{Cioni}, this rapid increase nicely fits with the predicted behavior, equation~\eqref{nure2}, in a GL case~\eqref{IV}\textsubscript{l}, and intermediate turbulence.

\begin{figure}[h!]
\includegraphics[width= 8cm]{Cioni}
\caption{Part of the Cioni \etal\ results~\cite{Cioni}, corresponding to their ``rapid increase'' in Nusselt number, in logarithmic coordinates. The continuous line has a slope 2.}\label{Cioni}
\end{figure}

This regime would be much better characterized if Cioni \etal\ could have measured the Reynolds number or/and the temperature fluctuations simultaneously with these observations. A behavior $\Nusselt \propto \Rayleigh^2$ is however sufficiently unusual. Moreover, just before this rapid increase, Cioni \etal\ observed a regime $\Nusselt \propto \Rayleigh^{1/5}$, which agrees with the low Prandtl number soft turbulence, equation~\eqref{nure6-}. The transition between this regime and the rapid increase of $\Nusselt$ would thus correspond to the transition soft to hard turbulence \ie\ the apparition of an inertial cascade. It strongly suggests that the Cioni \etal\ observation is the only known observed regime of type~\eqref{IV} in the GL classification.

\subsection{The observation of Chavanne \texorpdfstring{\etal}{et al.}}

Except the Cioni \etal\ observation discussed above, Chavanne \etal~\cite{Xav} were the first to observe a logarithmic slope $\mu$ of $\Nusselt$ versus $\Rayleigh$ larger than $1/3$. As a tentative interpretation, they assimilated this regime with the Kraichnan regime. However, the observed $\mu$ was smaller than $1/2$, which could be attributed to the logarithmic corrections. Later, other authors~\cite{He} observed a transition toward a regime with a similar value for $\mu$, larger than $1/3$ but smaller than $1/2$, while at much higher Rayleigh number.

Indeed, a logarithmic correction is hardly distinguishable from a power law with a small exponent, and even the power law observed on Figure~\ref{devlXav1} for the second regime could dissimulate a logarithmic correction. Let us however look at equations~\eqref{Bl} and~\eqref{K2}. They both give $\Nusselt$ as the product of a function of $\Reynolds$ and a power of $\Prandtl$. As $\ln(\Reynolds)$ mimics a power law, the behavior of the different functions of $\Reynolds$ should be close. For the final regime of Chavanne \etal, the exponent of $\Reynolds$, $1/(2-\alpha)\simeq 0.81$, and $\Nusselt/\Reynolds^{0.81}$ should be poorly dependent on $\Reynolds$ at constant $\Prandtl$.

We then select the data for which $3\cdot 10^5<\Reynolds<6\cdot 10^5$, and plot:
\begin{equation}
\frac{\Nusselt}{\Reynolds^{0.81}\Prandtl^{1/2}}
\end{equation}
versus the Prandtl number, in logarithmic coordinates (see Figure~\ref{Chav}). It would give a constant value if the original result of Kraichnan is the good candidate. It would give an ascending slope:
\begin{equation}
\frac{2}{3(2-\alpha)}-\frac{1}{2}\simeq 0.042
\end{equation}
in the Blasius case. The results seem to favor the last case.

Does the more recent results~\cite{He} correspond to the Blasius or the logarithmic case? It can only be decided through a similar test, involving a large range of Prandtl numbers.

\begin{figure}[h!]
\includegraphics[width= 8cm]{testBLogXav}
\caption{Dependence of the Nusselt number at ``constant'' Reynolds number ($3\cdot 10^5<\Reynolds<6\cdot 10^5$), for the Chavanne \etal\ results~\cite{Xav}. The constant dotted line corresponds to the Kraichnan~\cite{Kraichnan} solution. The continuous line corresponds to the Blasius boundary layer, with $\alpha=0.77$ (see text).}\label{Chav}
\end{figure}

\subsection{The Shishkina and Lohse model}

During the completion of this work, Shishkina and Lohse~\cite{SL} published a work which aims at clarifying the ``ultimate regimes''. They made a great step in this direction, being particularly careful to be consistent with previous~\cite{Doering} and new~\cite{Choffrut} exact heat transfer upper bounds. They distinguished four regimes, which they called~\eqref{IV}\textsuperscript{$\prime$}\textsubscript{l} and~\eqref{II}\textsuperscript{$\prime$}\textsubscript{l} for low Prandtl numbers, \eqref{IV}\textsuperscript{$\prime$}\textsubscript{u} and~\eqref{III}\textsuperscript{$\prime$}\textsubscript{$\infty$} for large ones.

While our paper is mainly concerned with the structure of the boundary layer, we also treated the bulk for completeness, and the relation between their results and ours is thus important to set. Let us begin with their regimes~\eqref{IV}\textsuperscript{$\prime$}\textsubscript{l} and~\eqref{IV}\textsuperscript{$\prime$}\textsubscript{u}. These regimes are dominated by the bulk thermal dissipation, and by the viscous dissipation in a logarithmic boundary layer. As a minor point, and while it is difficult to contradict one of the co-authors of the classification, we tend to consider that, logarithmic or not, a boundary layer is a boundary layer, and the classification would better be~\eqref{III}\textsuperscript{$\prime$}\textsubscript{l} and~\eqref{III}\textsuperscript{$\prime$}\textsubscript{u}. We are not in a channel flow, where the logarithmic boundary layer can invade the whole flow, and the exchange of momentum can occur between ascending plumes and descending ones, not only with the walls. This is indeed the case for convection in a vertical channel~\cite{Cholemari,Tisserand,Calzavarini}. We expect this is so at extreme Rayleigh numbers, as the non dimensional bulk viscous dissipation, equation~\eqref{bulk} grows more rapidly with $\Reynolds$ than the logarithmic boundary layer one, equation~\eqref{log}.

Our approach is different from the Shishkina and Lohse~\cite{SL} one. While our calculus is based on the exact relations concerning viscous and thermal dissipations (equations~\eqref{exactv} and~\eqref{exactt}), they prefer to modelize the turbulent velocity and heat diffusivities. Using a rather obscure hypothesis (between their equations~(11) and~(12)), they distinguish between low and large Prandtl numbers.

For low Prandtl numbers, we agree with Shishkina and Lohse~\cite{SL} on the viscous dissipation in the logarithmic boundary layer:
\begin{equation}\label{log2}
\Rayleigh \, \Nusselt=\frac{\Reynolds^3\Prandtl^2}{(\ln \Reynolds)^2},
\end{equation}
which is our equation~\eqref{log} and their equation~(16) with $\zeta=0$. For solving in $\Nusselt$ and $\Reynolds$, we use equations~\eqref{exactt} and~\eqref{dtbis}:
\begin{equation}
\frac{U\Th^2}{H}=\kappa{\parens[\Bigg]{\frac{\Delta}{H}}}^2\frac{UH}{\nu}\frac{\nu}{\kappa}\frac{\Th^2}{\Delta^2}=\kappa{\parens[\Bigg]{\frac{\Delta}{H}}}^2\Nusselt,
\end{equation}
and the evaluation of $\Theta/\Delta$ with a logarithmic boundary layer, equation~\eqref{thetalog}:
\begin{equation}
\frac{\Theta}{\Delta}=\frac{\Nusselt(\ln \Reynolds)}{\Reynolds \, \Prandtl}.
\end{equation}
It results in:
\begin{align}
\Reynolds&=\Rayleigh^{1/2}\Prandtl^{-1/2}, \\
\Nusselt&=\frac{\Rayleigh^{1/2}\Prandtl^{1/2}}{(\ln \Reynolds)^2},
\end{align}
in perfect agreement with Shishkina and Lohse~\cite{SL}.

Things are different for large Prandtl numbers. We consider that the viscous dissipation is not affected by the Prandtl number, and remain with our equation~\eqref{log2}. Their approach yields Shishkina and Lohse to introduce an additional $\Prandtl^{-1}$ factor:
\begin{equation}\label{logSL}
\Rayleigh \, \Nusselt=\frac{\Reynolds^3\Prandtl}{(\ln \Reynolds)^2}.
\end{equation}
It would mean that the viscous dissipation, out of the logarithmic correction, would not only depend on velocity and dimension of the flow, but also on the Prandtl number. This again seems in contradiction with the vertical channel convection~\cite{Cholemari,Tisserand,Calzavarini}.

As for us, we keep the previous equation~\eqref{log2}. The thermal bulk dissipation, as discussed in Section~\ref{thermalhpr}, is dominated by the boundary layer:
\begin{equation}\label{nuult+}
\Nusselt=\frac{H}{\delta}=\frac{\Reynolds \, \Prandtl^{1/3}}{\ln(\Reynolds)}.
\end{equation}
Using equations~\eqref{nuult+} and~\eqref{log2}, we obtain:
\begin{align}
\Reynolds&=\Rayleigh^{1/2}\Prandtl^{-5/6}(\ln \Reynolds)^{1/2}, \\
\Nusselt&=\frac{\Rayleigh^{1/2}\Prandtl^{-1/2}}{(\ln \Reynolds)^{1/2}}.\label{ivprime}
\end{align}
It differs from the predictions of Shishkina and Lohse~\cite{SL}:
\begin{align}
\Reynolds&=\Rayleigh^{1/2}\Prandtl^{-1/2}, \\
\Nusselt&=\frac{\Rayleigh^{1/2}\Prandtl^{-1/2}}{(\ln \Reynolds)^2},
\end{align}
both our predictions and theirs being consistent with the Doering and Constantin~\cite{Doering}, and Choffrut \etal~\cite{Choffrut} bounds.

For very high Prandtl numbers ($\Prandtl>\Rayleigh^{1/3}$), we agree with Shishkina and Lohse that there might be a transition toward $\Nusselt=\Rayleigh^{1/3}$. Indeed, the above predictions, equations~\eqref{ivprime} would imply $\Nusselt<\Rayleigh^{1/3}$, \ie\ a thermal boundary layer Rayleigh number, $\Rayleigh/\Nusselt^3$ indefinitely growing when $\Rayleigh$ grows, yielding to its instability.

However, as far as $\Prandtl<\Rayleigh^{2/3}$, the Reynolds number remains high. Neglecting the logarithmic corrections, equation~\eqref{bulk} holds:
\begin{equation}\label{iiiprime1}
\Rayleigh \, \Nusselt=\Reynolds^3\Prandtl^2= \Rayleigh^{4/3}.
\end{equation}
It gives:
\begin{align}
\Reynolds&=\Rayleigh^{4/9}\Prandtl^{-2/3},\\
\Nusselt&=\Rayleigh^{1/3}.\label{iiiprime}
\end{align}
Note that both our scalings and the SL ones ensure the continuity of $\Nusselt$ and $\Reynolds$ on the line $\Prandtl=\Rayleigh^{1/3}$.

For very low Prandtl numbers, Shishkina and Lohse~\cite{SL} estimate that a transition must occur from the regime~\eqref{IV}\textsuperscript{$\prime$}\textsubscript{l} to a regime they call~\eqref{II}\textsuperscript{$\prime$}\textsubscript{l}, along a line $\Rayleigh \, \Prandtl=\cst$. They justify this transition in the following term:
\begin{quote}
While moving along this line for an increasing $\Rayleigh$ and $\Prandtl \propto \Rayleigh^{-1}$, the Nusselt number remains constant, and any steeper transition slope from regime~\eqref{IV}\textsuperscript{$\prime$}\textsubscript{l} would imply an unphysical limit $\Nusselt \rightarrow 0$ along that line.
\end{quote}

But the same is true for the scaling they propose: $\Nusselt \propto (\Rayleigh \, \Prandtl)^{1/5}$. Indeed, they erroneously attribute to $\Nusselt$ the scaling of $(\Nusselt-1)$, as we show in the Section~\ref{vsp}.

Table~\ref{t5} compares our scalings to the SL ones, neglecting the logarithmic corrections.

%\begin{table}[ht!]
%\normalsize
%\centering
%\caption{\normalsize{Comparison between Shishkina and Lohse~\cite{SL} scalings and ours for ultimate regimes.}\label{t5}
%}
%\begin{tabular}{lll}
%\hline 	&&\\[-2.7ex]
%\hline	&&\\[-2ex]
%		& \textbf{Present work} & \textbf{Shishkina and Lohse~\cite{SL}} \\
%		&&\\[-2ex]
%\hline 	&&\\[-2.7ex]
%\hline	&&\\[-2ex]
%$\Prandtl<\Rayleigh^{-1}$ & $\Reynolds=\Rayleigh^{2/5}\Prandtl^{-3/5}$ & $\Reynolds=\Rayleigh^{2/5}\Prandtl^{-3/5}$ \\
%		& $(\Nusselt-1)=\Rayleigh^{1/5}\Prandtl^{1/5}$ & $\Nusselt=\Rayleigh^{1/5}\Prandtl^{1/5}$ \\
%		&&\\[-2ex]
%$\Rayleigh^{-1}<\Prandtl<1$ & $\Reynolds=\Rayleigh^{1/2}\Prandtl^{-1/2}$ & $\Reynolds=\Rayleigh^{1/2}\Prandtl^{-1/2}$ \\
%		&$\Nusselt=\Rayleigh^{1/2}\Prandtl^{1/2}$ & $\Nusselt=\Rayleigh^{1/2}\Prandtl^{1/2}$ \\
%		&&\\[-2ex]
%$1<\Prandtl<\Rayleigh^{1/3}$ &$\Reynolds=\Rayleigh^{1/2}\Prandtl^{-5/6}$ & $\Reynolds=\Rayleigh^{1/2}\Prandtl^{-1/2}$ \\
%&$\Nusselt=\Rayleigh^{1/2}\Prandtl^{-1/2}$ & $\Nusselt=\Rayleigh^{1/2}\Prandtl^{-1/2}$ \\
%		&&\\[-2ex]
%$\Rayleigh^{1/3}<\Prandtl<\Rayleigh^{2/3}$ & $\Reynolds=\Rayleigh^{4/9}\Prandtl^{-2/3}$ & $\Reynolds=\Rayleigh^{2/3}\Prandtl^{-1}$ \\
%&$\Nusselt=\Rayleigh^{1/3}$ & $\Nusselt=\Rayleigh^{1/3}$ \\
%		&&\\[-2ex]
%\hline 	&&\\[-2.7ex]
%\hline	&&\\[-2ex]
%\end{tabular}
%\end{table}
\begin{table}[h!]
\caption{Comparison between Shishkina and Lohse~\cite{SL} scalings and ours for ultimate regimes.\label{t5}}
\begin{tabular}{lll}
\hline 	&&\\[-2.7ex]
\hline	&&\\[-2ex]
		& \textbf{Present work} & \textbf{Shishkina and Lohse~\cite{SL}} \\
		&&\\[-2ex]
\hline 	&&\\[-2.7ex]
\hline	&&\\[-2ex]
$\Prandtl<\Rayleigh^{-1}$
 		& 	\begin{tabular}{@{}l@{}}
			$\Reynolds=\Rayleigh^{2/5}\Prandtl^{-3/5}$
		\\	$(\Nusselt-1)=\Rayleigh^{1/5}\Prandtl^{1/5}$
			\end{tabular}
		& 	\begin{tabular}{@{}l@{}}
			$\Reynolds=\Rayleigh^{2/5}\Prandtl^{-3/5}$
		\\	$\Nusselt=\Rayleigh^{1/5}\Prandtl^{1/5}$
			\end{tabular}\\
		&&\\[-2ex]
$\Rayleigh^{-1}<\Prandtl<1$
		& 	\begin{tabular}{@{}l@{}}
			$\Reynolds=\Rayleigh^{1/2}\Prandtl^{-1/2}$
		\\	$\Nusselt=\Rayleigh^{1/2}\Prandtl^{1/2}$
			\end{tabular}
		& 	\begin{tabular}{@{}l@{}}
			$\Reynolds=\Rayleigh^{1/2}\Prandtl^{-1/2}$
		\\ 	$\Nusselt=\Rayleigh^{1/2}\Prandtl^{1/2}$
			\end{tabular}\\
		&&\\[-2ex]
$1<\Prandtl<\Rayleigh^{1/3}$
		& 	\begin{tabular}{@{}l@{}}
			$\Reynolds=\Rayleigh^{1/2}\Prandtl^{-5/6}$ 
		\\	$\Nusselt=\Rayleigh^{1/2}\Prandtl^{-1/2}$
			\end{tabular}
		& 	\begin{tabular}{@{}l@{}}
			$\Reynolds=\Rayleigh^{1/2}\Prandtl^{-1/2}$
		\\	$\Nusselt=\Rayleigh^{1/2}\Prandtl^{-1/2}$
			\end{tabular}\\
		&&\\[-2ex]
$\Rayleigh^{1/3}<\Prandtl<\Rayleigh^{2/3}$
		& 	\begin{tabular}{@{}l@{}}
			$\Reynolds=\Rayleigh^{4/9}\Prandtl^{-2/3}$
		\\	$\Nusselt=\Rayleigh^{1/3}$
			\end{tabular}
		& 	\begin{tabular}{@{}l@{}}
			$\Reynolds=\Rayleigh^{2/3}\Prandtl^{-1}$
		\\	$\Nusselt=\Rayleigh^{1/3}$
			\end{tabular}\\
		&&\\[-2ex]
\hline 	&&\\[-2.7ex]
\hline
\end{tabular}
\end{table}

\section{Conclusion}\label{conc}

Let us first conclude on the comparison between this work and the results of Shishkina and Lohse~\cite{SL}. For moderately low Prandtl numbers (regime~\eqref{IV}\textsuperscript{$\prime$}\textsubscript{l}), we perfectly agree, except that, in our opinion, the really ``ultimate'' regime is free of logarithmic corrections. For extremely low Prandtl numbers (regime~\eqref{II}\textsuperscript{$\prime$}\textsubscript{l}), they erroneously attribute to $\Nusselt$ the scaling of $(\Nusselt-1)$, but we think this is a misprint due to a lack of attention.

For high Prandtl numbers (regimes~\eqref{IV}\textsuperscript{$\prime$}\textsubscript{u} and~\eqref{III}\textsuperscript{$\prime$}\textsubscript{u}), we disagree. Both our scaling and theirs fit the Constantin--Doering~\cite{Doering} and Choffrut \etal~\cite{Choffrut} bounds, and both propose a continuous transition, for $\Nusselt$ and $\Reynolds$, on the line $\Prandtl=\Rayleigh^{1/3}$. Nevertheless, the scalings proposed by Shishkina and Lohse~\cite{SL} do not agree with the common knowledge on turbulence: their turbulent viscous dissipation is sensitive to the Prandtl number.

Let us give a further argument against the unusual viscous dissipation proposed by Shishkina and Lohse (equation~\eqref{logSL}). The free fall velocity of plumes of temperature $\Theta=\Delta \Nusselt/\Reynolds \, \Prandtl$ is $\sqrt{g\beta\Theta H}$, and the corresponding Reynolds number $\Reynolds_{\ff}$ is given by:
\begin{equation}\label{ff1}
\Reynolds_{\ff}^2=\frac{g\beta\Theta H}{\nu^2}H^2=\frac{\Rayleigh}{\Prandtl}\frac{\Nusselt}{\Reynolds \, \Prandtl}.
\end{equation}
Thus, following the Shishkina and Lohse formula, equation~\eqref{logSL}:
\begin{equation}\label{ff2}
\Reynolds_{\ff}^2=\frac{\Rayleigh \, \Nusselt}{\Reynolds \, \Prandtl^2}=\frac{\Reynolds^2}{(\ln \Reynolds)^2\Prandtl},
\end{equation}
which means that the free fall velocity of the plumes is smaller than the typical velocity of the flow. Our scalings avoid this drawback, but are deeply inspired by their study.

However, the present paper is mainly concerned with the structure of the boundary layer, and its consequences on the dependence of Nusselt and Reynolds numbers on Rayleigh and Prandtl numbers. We succeed in taking into account an often made observation of plumes on the plates, and gave a precise interpretation of Puthenveettil \etal~\cite{PA} measurements of the total length of plumes per unit area. We show how this total length, or equivalently the distance between plumes, can be related to the observed $\Nusselt$ and $\Reynolds$ numbers. The relation between this distance between plumes and the Nusselt number is demonstrated both experimentally and theoretically. The theoretical model we used assumes that the heat flux can be correctly estimated through the product of typical velocity and typical temperature fluctuation. A critical verification of this hypothesis showed us that it is not always the case.

Thus, according to the present work, the various Rayleigh--B\'enard convection regimes can be separated in two kinds:
\begin{itemize}
\item The unscaling ones. Either because of a transition between two scaling regimes, or due to a special organization of the flow to be elucidated, the cross-correlation coefficient between velocity and temperature cannot be considered as constant. A fundamental hypothesis of many of the proposed models then fails. One can then only rely to the GL theory~\cite{GL}. Even if their estimation of the boundary layer viscous dissipation appears as too crude, the more elaborated evaluations, like equation~\eqref{epsBla}, are always intermediate between the crude estimation and the bulk dissipation. Thus, estimating the total dissipation as a pondered mean of these two extremes should correctly capture the physics.
\item The scaling ones. Then, the cross-correlation coefficient between velocity and temperature can be considered as constant, and a deeper analysis can be performed. It reveals that different regimes are possible, even for the same Rayleigh number $\Rayleigh$, Prandtl number $\Prandtl$, and aspect ratio $\Gamma$. It also reveals that the evolution of the viscous boundary layer proceeds through a succession of steps toward the turbulent state, rather than through an abrupt and single transition.
\end{itemize}

One could wonder why we mainly used relatively old experimental results, while more recent, often more precise works exist~\cite{He,Niemela,Roche}, including numerical ones~\cite{Verzicco,Schumacher}. The reason is that these works do not provide simultaneous measurements of $\Nusselt$, $\Reynolds$ and $\Th$. Different regimes have been observed, at the same Rayleigh and Prandtl numbers, even in the same cell. So, simultaneous measurements are essentials. Except in very special cases, as the rapid increase observed by Cioni \etal~\cite{Cioni}, the behavior of the Nusselt number alone cannot allow to conclude.

The numerical works have the advantage to give access to all the possible measurements. However, for obvious reasons, they hardly give dense series in the $\Rayleigh$, $\Prandtl$ plane. As different regimes rapidly succeed each other, isolated measurements again cannot allow to conclude.

Let us enumerate some aspects of Rayleigh--B\'enard convection on which our study gives a new point of view:
\begin{itemize}
\item Several studies on two dimensional (2D) Rayleigh--B\'enard convection have pointed the remarkable similitude with three dimensions~(3D). Indeed, up to the Soft Turbulence included, there are no real differences between 2D and 3D. Only in the Hard Turbulence regime, the energy or enstrophy cascades are dramatically different in 2D and in 3D. In particular, equations~\eqref{exactv} and~\eqref{exactt} are valid in 2D. Moreover, the influence of the energy cascade only appears in the bulk thermal dissipation, which never dominates in the large Prandtl numbers ultimate regimes. It suggests that a difference between 2D and 3D can only occur at low Prandtl numbers.
\item Sharp transitions in the cell behavior can result either from a transition in the bulk (as the transition soft turbulence to hard turbulence) or from a change in the structure of the boundary layer. The latter is not limited to the laminar to turbulent transition. Successive changes in the $\alpha$ value are steps in the evolution from laminar to turbulent. This is exactly the conclusion of Gauthier \etal~\cite{Gauthier} about the Chavanne observation. They observe that, after the transition, the spectrum of plate temperature fluctuations extends to much higher frequencies, which is coherent with a much smaller development length $h$.
\item Temperature fluctuations measurements are generally easier and more reliable than velocity measurements. Assuming the constancy of the cross-correlation coefficient between vertical velocity and temperature (an assumption implicitly made in several models) gives indirectly access to the Reynolds number through temperature measurements. We urge all experimentalists to include these temperature fluctuations in their data.
\end{itemize}

\section*{Acknowledgments}

The authors are thankful to Marc Moulin and his team at the mechanical workshop for the design and machining of the experimental apparatus. We thank Vera Musilov\'a, Michal Macek and Pavel Urban for sharing their data table with us.

\section*{Declaration of interests}

The authors do not work for, advise, own shares in, or receive funds from any organization that could benefit from this article, and have declared no affiliations other than their research organizations.

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